Review

An Alternative to Domain-general or Domain-specific Frameworks for Theorizing about Human Evolution and Ontogenesis

  • Received: 01 January 2015 Accepted: 11 May 2015 Published: 19 June 2015
  • This paper maintains that neither a domain-general nor a domain-specific framework is appropriate for furthering our understanding of human evolution and ontogenesis. Rather, as we learn increasingly more about the dynamics of gene-environment interaction and gene expression, theorists should consider a third alternative: a domain-relevant approach, which argues that the infant brain comes equipped with biases that are relevant to, but not initially specific to, processing different kinds of input. The hypothesis developed here is that domain-specific core knowledge/specialized functions do not constitute the start state; rather, functional specialization emerges progressively through neuronal competition over developmental time. Thus, the existence of category-specific deficits in brain-damaged adults cannot be used to bolster claims that category-specific or domain-specific modules underpin early development, because neural specificity in the adult brain is likely to have been the emergent property over time of a developing, self-structuring system in interaction with the environment.

    Citation: Annette Karmiloff-Smith. An Alternative to Domain-general or Domain-specific Frameworks for Theorizing about Human Evolution and Ontogenesis[J]. AIMS Neuroscience, 2015, 2(2): 91-104. doi: 10.3934/Neuroscience.2015.2.91

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  • This paper maintains that neither a domain-general nor a domain-specific framework is appropriate for furthering our understanding of human evolution and ontogenesis. Rather, as we learn increasingly more about the dynamics of gene-environment interaction and gene expression, theorists should consider a third alternative: a domain-relevant approach, which argues that the infant brain comes equipped with biases that are relevant to, but not initially specific to, processing different kinds of input. The hypothesis developed here is that domain-specific core knowledge/specialized functions do not constitute the start state; rather, functional specialization emerges progressively through neuronal competition over developmental time. Thus, the existence of category-specific deficits in brain-damaged adults cannot be used to bolster claims that category-specific or domain-specific modules underpin early development, because neural specificity in the adult brain is likely to have been the emergent property over time of a developing, self-structuring system in interaction with the environment.


    Let $ B $ be a subset of a topological space $ \left(Y, \sigma \right) $. The set of condensation points of $ B $ is denoted by $ Cond\left(B\right) $ and defined by $ Cond\left(B\right) = \left\{ y\in Y\text{: for every }V\in \sigma \text{ with }y\in V\text{, }V\cap B\text{ is uncountable}\right\} $. For the purpose of characterizing Lindelöf topological spaces and improving some known mapping theorems, the author in [1] defined $ \omega $-closed sets as follows: $ B $ is called an $ \omega $-closed set in $ \left(Y, \sigma \right) $ if $ Cond\left(B\right) \subseteq B $. $ B $ is called an $ \omega $-open set in $ \left(Y, \sigma \right) $ [1] if $ Y-B $ is $ \omega $-closed. It is well known that $ B $ is $ \omega $-open in $ \left(Y, \sigma \right) $ if and only if for every $ b\in B $, there are $ V\in \sigma $ and a countable subset $ F\subseteq Y $ with $ b\in V-F\subseteq B $. The family of all $ \omega $-open sets in $ \left(Y, \sigma \right) $ is denoted by $ \sigma _{\omega } $. It is well known that $ \sigma _{\omega } $ is a topology on $ Y $ that is finer than $ \sigma $. Many research papers related to $ \omega $-open sets have appeared in [2,3,4,5,6,7,8] and others. Authors in [9,10,11] included $ \omega $-openness in both soft and fuzzy topological spaces. In this paper, we will denote the closure of $ B $ in $ \left(Y, \sigma \right) $, the closure of $ B $ in $ \left(Y, \sigma _{\omega }\right) $, the interior of $ B $ in $ \left(Y, \sigma \right) $, and the interior of $ B $ in $ \left(Y, \sigma _{\omega }\right) $ by $ \overline{B} $, $ \overline{B}^{\omega } $, $ Int\left(B\right) $, and $ Int_{\omega }\left(B\right) $, respectively. $ B $ is called a semi-open set in $ \left(Y, \sigma \right) $ [12] if there exists $ V\in \sigma $ such that $ V\subseteq B\subseteq \overline{V} $. Complements of semi-open sets are called semi-closed sets. The family of all semi-open sets in $ \left(Y, \sigma \right) $ will be denoted by $ SO\left(Y, \sigma \right) $. Authors in [12,13,14,15] have used semi-open sets to define semi-continuity, semi-openness, irresoluteness, pre-semi-openness, and slight semi-continuity. The area of research related to semi-open sets is still hot [16,17,18,19,20,21,22,23,24,25,26,27,28]. Authors in [29] have defined $ \omega _{s} $-open sets as a strong form of semi-open sets as follows: $ B $ is called an $ \omega _{s} $-open set in $ \left(Y, \sigma \right) $ if there exists $ O\in \sigma $ such that $ O\subseteq B\subseteq \overline{O}^{\omega } $. Complements of $ \omega _{s} $-open sets are called $ \omega _{s} $-closed sets. The family of all $ \omega _{s} $-open sets in $ \left(Y, \sigma \right) $ will be denoted by $ \omega _{s}\left(Y, \sigma \right) $. The intersection of all $ \omega _{s} $-closed sets in $ \left(Y, \sigma \right) $ which contains $ B $ will be denoted by $ \overline{B} ^{\omega _{s}} $, and the union of all $ \omega _{s} $-open sets in $ \left(Y, \sigma \right) $ which contained in $ B $ will be denoted by $ Int_{\omega _{s}}\left(B\right) $. Authors in [29] have defined and investigated the class of $ \omega _{s} $-continuity which lies strictly between the classes of continuity and semi-continuity.

    In this paper, $ \omega _{s} $-irresoluteness as a strong form of $ \omega _{s} $-continuity is introduced. It is proved that $ \omega _{s} $-irresoluteness is independent of each of continuity and $ \omega _{s} $-irresoluteness. Also, $ \omega _{s} $-openness which lies strictly between openness and semi-openness is introduced and investigated, and pre-$ \omega _{s} $-openness which is a strong form of $ \omega _{s} $ -openness and independent of openness is introduced and investigated. Moreover, slight $ \omega _{s} $-continuity as a new class of functions which lies between slight continuity and slightly semi-continuity is introduced and investigated. In addition to these, $ \omega _{s} $-compactness as a new class of topological spaces that lies strictly between compactness and semi-compactness is introduced. Several implications, examples, counter-examples, characterizations, and mapping theorems are introduced. In particular, several sufficient conditions for the equivalence between our new concepts and other related concepts are given. We hope that this will open the door for a number of future related studies such as $ \omega _{s} $ -separation axioms and $ \omega _{s} $-connectedness.

    Throughout this paper, the usual topology on $ \mathbb{R} $ will be denoted by $ \tau _{u} $.

    Recall that a topological space $ (Y, \sigma) $ is called locally countable [30] (resp. anti-locally countable [31]) if $ (Y, \sigma) $ has a base consisting of countable sets (all non-empty open sets are uncountable sets).

    The following results will be used in the sequel:

    Proposition 1.1. [29] Let $ (Y, \sigma) $ be a topological space. Then

    (a) $ \sigma \subseteq \omega _{s}(Y, \sigma)\subseteq SO(Y, \sigma) $.

    (b) If $ (Y, \sigma) $ is locally countable, then $ \sigma = \omega _{s}(Y, \sigma) $.

    (c) If $ (Y, \sigma) $ is anti-locally countable, then $ \omega _{s}(Y, \sigma) = SO(Y, \sigma) $.

    Proposition 1.2. [29] Let $ (Y, \sigma) $ be a topological space and let $ B, C\subseteq Y $. Then we have the following:

    (a) If $ B\subseteq C\subseteq \; \overline{B}^{\omega } $ and $ B \; \in \; \omega _{s}(Y, \sigma) $, then $ C \; \in \omega _{s}(Y, \sigma) $.

    (b) If $ B\in \sigma $ and $ C \; \in \omega _{s}(Y, \sigma) $, then $ B \; \cap \; C\in \omega _{s}(Y, \sigma) $.

    Proposition 1.3. [31] Let $ (Y, \sigma) $ be anti-locally countable. Then for every $ B \; \in \sigma _{\omega } $, we have $ \overline{B}^{\omega } = \overline{B} $.

    Definition 2.1. A function $ g:(Y, \sigma)\longrightarrow \left(Z, \gamma \right) $ is called irresolute [14] (resp. $ \omega _{s} $ -continuous [29]), if for every $ V\in SO\left(Z, \gamma \right) $ (resp. $ V\in \gamma $), $ g^{-1}(V)\in SO(Y, \sigma) $ (resp. $ g^{-1}(V)\in \omega _{s}(Y, \sigma) $).

    Definition 2.2. A function $ g:(Y, \sigma)\longrightarrow \left(Z, \gamma \right) $ is called $ \omega _{s} $-irresolute, if for every $ V\in \omega _{s}\left(Z, \gamma \right) $, $ g^{-1}(V)\in \omega _{s}(Y, \sigma) $.

    The following two examples will show that irresoluteness and $ \omega _{s} $ -irresoluteness are independent concepts:

    Example 2.3. Let $ X = \mathbb{R} $, $ Y = \left\{ a, b\right\} $, $ \tau = \left\{ \emptyset, \mathbb{R}, \mathbb{N}, \mathbb{Q} ^{c}, \mathbb{N} \cup \mathbb{Q} ^{c}\right\} $, and $ \sigma = \left\{ \emptyset, Y, \left\{ a\right\}, \left\{ b\right\} \right\} $. Define $ f:(X, \tau)\longrightarrow (Y, \sigma) $ by

    $  f(x)={aif xQcbif xQ.
    $

    Since $ f^{-1}\left(\left\{ a\right\} \right) = \mathbb{Q} ^{c}\in \tau \subseteq SO(X, \tau) $ and $ f^{-1}\left(\left\{ b\right\} \right) = \mathbb{Q} \in SO(X, \tau)-\omega _{s}(X, \tau) $, then $ f $ is irresolute but not $ \omega _{s} $-irresolute.

    Example 2.4. Consider the topology $ \sigma = \left\{ \emptyset, \mathbb{N}, \left\{ 1\right\}, \left\{ 2\right\}, \left\{ 1, 2\right\} \right\} $ on $ \mathbb{N} $. Define $ g:(\mathbb{N}, \sigma)\longrightarrow (\mathbb{N}, \sigma) $ by

    $ g(t)={1if t=11if t=2tif tN{1,2}.
    $

    Since $ (\mathbb{N}, \sigma) $ is locally countable, then by Proposition 1.1 (b) we have $ \omega _{s}(\mathbb{N}, \sigma) = \sigma $. Since $ f^{-1}\left(\left\{ 1\right\} \right) = \left\{ 1, 2\right\} \in \sigma $ and $ f^{-1}\left(\left\{ 2\right\} \right) = \emptyset \in \sigma $, then $ f $ is $ \omega _{s} $-irresolute. However, $ f $ is not irresolute because there is $ \overline{\left\{ 2\right\} } = \mathbb{N} -\left\{ 1\right\} \in SO(\mathbb{N}, \sigma) $ such that $ f^{-1}\left(\mathbb{N} -\left\{ 1\right\} \right) = \mathbb{N} -\left\{ 1, 2\right\} \notin SO(\mathbb{N}, \sigma) $.

    Theorem 2.5. Let $ \left(Y, \sigma \right) $ and $ (Z, \gamma) $ be two anti-locally countable topological spaces. Then for any function $ g:\left(Y, \sigma \right) \longrightarrow (Z, \gamma) $ the followings are equivalent:

    (a) $ g $ is irresolute.

    (b) $ g $ is $ \omega _{s} $-irresolute.

    Proof. Follows from the definitions and Proposition 1.1 (c).

    The following two examples will show that continuity and $ \omega _{s} $ -irresoluteness are independent concepts:

    Example 2.6. Let $ X = \mathbb{R} \ $and $ \tau = \left\{ \emptyset, \mathbb{R}, \left(3, \infty \right), \left\{ 2\right\}, \left\{ 2\right\} \cup \left(3, \infty \right) \right\} $. Define $ f:(X, \tau)\longrightarrow (X, \tau) $ by

    $ f(x)={2if x{2}(3,)xif xR({2}(3,)).
    $

    Since $ f^{-1}\left(\left(3, \infty \right) \right) = \emptyset \in \tau $ and $ f^{-1}\left(\left\{ 2\right\} \right) = \left\{ 2\right\} \cup \left(3, \infty \right) \in \tau $, then $ f $ is continuous. Since $ \overline{\left(3, \infty \right) }^{\omega } = \mathbb{R} -\left\{ 2\right\} $, then $ \mathbb{R} -\left\{ 2\right\} \in \omega _{s}(X, \tau) $. Since $ f^{-1}\left(\mathbb{R} -\left\{ 2\right\} \right) = \mathbb{R} -\left(\left\{ 2\right\} \cup \left(3, \infty \right) \right) \notin \omega _{s}(X, \tau) $, then $ f $ is not $ \omega _{s} $-irresolute.

    Example 2.7. Let $ \tau _{disc} $ be the discrete topology on $ \left\{ a, b\right\} $. Define $ f:(\mathbb{R}, \tau _{u})\longrightarrow (\left\{ a, b\right\}, \tau _{disc}) $ by

    $ f(x)={aif x(,0)bif x[0,).
    $

    Clearly that $ \omega _{s}(\left\{ a, b\right\}, \tau _{disc}) = \tau _{disc} $. Since $ f^{-1}\left(\left\{ a\right\} \right) = \left(-\infty, 0\right) \in \tau _{u}\subseteq \omega _{s}(\mathbb{R}, \tau _{u}) $ and $ f^{-1}\left(\left\{ b\right\} \right) = \left[0, \infty \right) \in \omega _{s}(\mathbb{R}, \tau _{u}) $, then $ f $ is $ \omega _{s} $-irresolute. However, $ f $ is not continuous because there is $ \left\{ b\right\} \in \tau _{disc} $ such that $ f^{-1}\left(\left\{ b\right\} \right) = \left[0, \infty \right) \notin \tau _{u} $.

    Theorem 2.8. Let $ \left(Y, \sigma \right) $ and $ (Z, \gamma) $ be two locally countable topological spaces. Then for any function $ g:\left(Y, \sigma \right) \longrightarrow (Z, \gamma) $ the followings are equivalent:

    (a) $ g $ is continuous.

    (b) $ g $ is $ \omega _{s} $-irresolute.

    Proof. Follows from the definitions and Proposition 1.1 (b).

    Theorem 2.9. If $ g:\left(Y, \sigma \right) \longrightarrow (Z, \gamma) $ is a continuous function such that $ g:\left(Y, \sigma _{\omega }\right) \longrightarrow (Z, \gamma _{\omega }) $ is an open function, then $ g $ is $ \omega _{s} $-irresolute.

    Proof. Let $ H\in \omega _{s}(Z, \gamma) $. Then there exists $ W\in \gamma $ such that $ W\subseteq H\subseteq \overline{W}^{\omega } $ and so $ g^{-1}\left(W\right) \subseteq g^{-1}\left(H\right) \subseteq g^{-1}\left(\overline{W}^{\omega }\right) $. Since $ g:\left(Y, \sigma \right) \longrightarrow (Z, \gamma) $ is continuous, then $ g^{-1}(W)\in \sigma $. Since $ g:\left(Y, \sigma _{\omega }\right) \longrightarrow (Z, \gamma _{\omega }) $ is open, then $ g^{-1}\left(\overline{W}^{\omega }\right) \subseteq \overline{g^{-1}\left(W\right) }^{\omega } $. Thus, $ g^{-1}\left(W\right) \subseteq g^{-1}\left(H\right) \subseteq \overline{g^{-1}\left(W\right) }^{\omega } $, and hence $ g^{-1}\left(H\right) \in \omega _{s}\left(Y, \sigma \right) $. Therefore, $ g $ is $ \omega _{s} $-irresolute.

    Theorem 2.10. Every $ \omega _{s} $-irresolute function is $ \omega _{s} $-continuous.

    Proof. Let $ f:(X, \tau)\longrightarrow (Y, \sigma) $ be $ \omega _{s} $ -irresolute. Let $ V\in \sigma $. Then by Proposition 1.1 (a), $ V\in \omega _{s}(Y, \sigma) $. Since $ f $ is $ \omega _{s} $-irresolute, then $ f^{-1}\left(V\right) \in \omega _{s}(X, \tau) $. Therefore, $ f $ is $ \omega _{s} $ -irresolute.

    The function in Example 2.6 is continuous and hence $ \omega _{s} $ -continuous. Therefore, the converse of Theorem 2.10 is not true, in general.

    Theorem 2.11. A function $ g:\left(Y, \sigma \right) \longrightarrow (Z, \gamma) $ is $ \omega _{s} $-irresolute if and only if for every $ \omega _{s} $-closed subset $ B $ of $ (Z, \gamma) $, $ g^{-1}\left(B\right) $ is $ \omega _{s} $-closed in $ \left(Y, \sigma \right) $.

    Proof. Necessity. Assume that $ g $ is $ \omega _{s} $-irresolute. Let $ B $ be an $ \omega _{s} $-closed set in $ (Z, \gamma) $. Then $ Z-B\in \omega _{s}(Z, \gamma) $. Since $ g $ is $ \omega _{s} $-irresolute, then $ g^{-1}\left(Z-B\right) = Y-g^{-1}\left(B\right) \in \omega _{s}\left(Y, \sigma \right) $. Hence, $ g^{-1}\left(B\right) $ is $ \omega _{s} $-closed in $ \left(Y, \sigma \right) $.

    Sufficiency. Suppose that for every $ \omega _{s} $-closed subset $ B $ of $ (Z, \gamma) $, $ g^{-1}\left(B\right) $ is $ \omega _{s} $ -closed in $ \left(Y, \sigma \right) $. Let $ W\in \omega _{s}(Z, \gamma) $. Then $ Z-W $ is $ \omega _{s} $-closed in $ (Z, \gamma) $. By assumption, $ g^{-1}\left(Z-W\right) = Y-g^{-1}\left(W\right) $ is $ \omega _{s} $-closed in $ \left(Y, \sigma \right) $, and so $ g^{-1}\left(W\right) \in \omega _{s}\left(Y, \sigma \right) $. Therefore, $ g $ is $ \omega _{s} $-irresolute.

    Theorem 2.12. A function $ g:\left(Y, \sigma \right) \longrightarrow (Z, \gamma) $ is $ \omega _{s} $-irresolute if and only if for every $ H\subseteq Y $, $ g\left(\overline{H}^{\omega _{s}}\right) \subseteq \overline{g\left(H\right) }^{\omega _{s}} $.

    Proof. Necessity. Suppose that $ g $ is $ \omega _{s} $-irresolute and let $ H\subseteq Y $. Then $ \overline{g\left(H\right) } ^{\omega _{s}} $ is $ \omega _{s} $-closed in $ (Z, \gamma) $, and by Theorem 2.11, $ g^{-1}\left(\overline{g\left(H\right) }^{\omega _{s}}\right) $ is $ \omega _{s} $-closed in $ \left(Y, \sigma \right) $. Since $ H\subseteq g^{-1}\left(\overline{g\left(H\right) }^{\omega _{s}}\right) $, then $ \overline{H}^{\omega _{s}}\subseteq g^{-1}\left(\overline{g\left(A\right) } ^{\omega _{s}}\right) $. Thus,

    $ g(¯Hωs)g(g1(¯g(H)ωs))¯g(H)ωs.
    $

    Sufficiency. Suppose that for every subset $ H\subseteq Y $, $ g\left(\overline{H}^{\omega _{s}}\right) \subseteq \overline{g\left(H\right) } ^{\omega _{s}} $. We will apply Theorem 2.11 to show that $ g $ is $ \omega _{s} $ -irresolute. Let $ B $ be an $ \omega _{s} $-closed subset of $ (Z, \gamma) $. Then by assumption we have $ g\left(\overline{g^{-1}(B)}^{\omega _{s}}\right) \subseteq \overline{g\left(g^{-1}(B\right))}^{\omega _{s}}\subseteq \overline{B}^{\omega _{s}} = B $, and so

    $ ¯g1(B)ωsg1(g(¯g1(B)ωs))g1(B).
    $

    Therefore, $ \overline{g^{-1}(B)}^{\omega _{s}} = g^{-1}\left(B\right) $, and hence $ g^{-1}\left(B\right) $ is $ \omega _{s} $-closed in $ \left(Y, \sigma \right) $. This shows that $ g $ is $ \omega _{s} $-irresolute.

    Theorem 2.13. A function $ g:\left(Y, \sigma \right) \longrightarrow (Z, \gamma) $ is $ \omega _{s} $-irresolute if and only if for every $ H\subseteq Z $, $ \overline{g^{-1}\left(H\right) }^{\omega _{s}}\subseteq g^{-1}\left(\overline{H}^{\omega _{s}}\right) $.

    Proof. Necessity. Suppose that $ g $ is $ \omega _{s} $-irresolute and let $ H\subseteq Z $. Then by Theorem 2.11, $ g^{-1}\left(\overline{H}^{\omega _{s}}\right) $ is $ \omega _{s} $-closed in $ \left(Y, \sigma \right) $. Since $ g^{-1}\left(H\right) \subseteq g^{-1}\left(\overline{H}^{\omega _{s}}\right) $, then $ \overline{g^{-1}\left(H\right) } ^{\omega _{s}}\subseteq g^{-1}\left(\overline{H}^{\omega _{s}}\right) $.

    Sufficiency. Suppose that for every $ H\subseteq Z $, $ \overline{ g^{-1}\left(H\right) }^{\omega _{s}}\subseteq g^{-1}\left(\overline{H} ^{\omega _{s}}\right) $. We will apply Theorem 2.11 to show that $ g $ is $ \omega _{s} $-irresolute. Let $ B $ be an $ \omega _{s} $-closed subset of $ (Z, \gamma) $. Then $ \overline{B}^{\omega _{s}} = B $. So by assumption, $ \overline{g^{-1}(B)}^{\omega _{s}}\subseteq g^{-1}\left(B\right) $, and so $ \overline{g^{-1}(B)}^{\omega _{s}} = g^{-1}\left(B\right) $. Therefore, $ g^{-1}\left(B\right) $ is $ \omega _{s} $-closed in $ \left(Y, \sigma \right) $.

    Theorem 2.14. The composition of two $ \omega _{s} $ -irresolute functions is $ \omega _{s} $-irresolute.

    Proof. Let $ g:(Y, \sigma)\longrightarrow (Z, \lambda) $ and $ h:(Z, \lambda)\longrightarrow (M, \gamma) $ be $ \omega _{s} $-irresolute functions. Let $ C\in \omega _{s}(M, \gamma) $. Since $ h $ is $ \omega _{s} $ -irresolute, then $ h^{-1}(C)\in \omega _{s}(Z, \lambda) $. Since $ g $ is $ \omega _{s} $-irresolute, then $ (h\circ g)^{-1}(C) = g^{-1}\left(h^{-1}(C)\right) \in \omega _{s}(Y, \sigma) $. Therefore, $ h\circ g $ is $ \omega _{s} $-irresolute.

    Definition 3.1. A function $ g:(Y, \sigma)\longrightarrow (Z, \lambda) $ is called

    (a) $ \omega _{s} $-open (resp. semi-open [13]) if for each $ U\in \sigma $, $ f(U)\in \omega _{s}(Z, \lambda) $ (resp. $ f(U)\in SO(Z, \lambda) $).

    (b) pre-semi-open [14] if for each $ U\in SO(Y, \sigma) $, $ f(U)\in SO(Z, \lambda) $.

    Theorem 3.2. Let $ g:(Y, \sigma)\longrightarrow (Z, \lambda) $ be a function. If for a base $ \mathcal{B} $ of $ (Y, \sigma) $, $ g(B)\in \lambda $ for all $ B\in \mathcal{B} $, then $ g $ is $ \omega _{s} $-open.

    Proof. Suppose that for a base $ \mathcal{B} $ of $ (Y, \sigma) $, $ g(B)\in \omega _{s}(Z, \lambda) $ for all $ B\in \mathcal{B} $. Let $ V\in \sigma -\left\{ \emptyset \right\} $. Choose $ \mathcal{B}_{1}\subseteq \mathcal{B} $ such that $ V = \cup \left\{ B:B\in \mathcal{B}_{1}\right\} $. Then

    $ g(V)=g({B:BB1})={g(B):BB1}.
    $

    Since by assumption, $ g\left(B\right) \in \omega _{s}(Z, \lambda) $ for all $ B\in \mathcal{B}_{1} $, then $ g\left(V\right) \in \omega _{s}(Z, \lambda) $.

    Theorem 3.3. Every open function is $ \omega _{s} $-open.

    Proof. Let $ g:\left(Y, \sigma \right) \longrightarrow (Z, \gamma) $ be an open function and let $ V\in \sigma $. Since $ g $ is open, then $ g(V)\in \gamma \subseteq \omega _{s}(Z, \gamma) $.

    The converse of Theorem 3.3 is not true as shown in the next example:

    Example 3.4. Consider the function $ g:(\mathbb{R}, \tau _{u})\longrightarrow (\mathbb{R}, \tau _{u}) $ defined by $ g\left(y\right) = y^{2} $. We apply Theorem 3.2 to show that $ g $ is $ \omega _{s} $-open. Consider the base $ \left\{ \left(c, d\right) :c, d\in \mathbb{R} \text{ and }c < d\right\} $ for $ (\mathbb{R}, \tau _{u}) $. Then for all $ c, d\in \mathbb{R} $ with $ c < d $ we have

    $ g((c,d))={(d2,c2)if c<d0(c2,d2)if 0c<d[0,max{c2,d2})if c<0<d
    $

    and so $ g\left(\left(c, d\right) \right) \in \omega _{s}(\mathbb{R}, \tau _{u}) $. Therefore, $ g $ is $ \omega _{s} $-open. On the other hand, since $ \mathbb{R} \in \tau _{u} $ but $ g\left(\mathbb{R} \right) = \left[0, \infty \right) \notin \tau _{u} $, then $ g $ is not open.

    Theorem 3.5. If $ g:\left(Y, \sigma \right) \longrightarrow (Z, \gamma) $ is an $ \omega _{s} $-open function such that $ (Z, \gamma) $ is locally countable, then $ g $ is open.

    Proof. Follows from the definitions and Proposition 1.1 (b).

    Theorem 3.6. Every $ \omega _{s} $-open function is semi-open.

    Proof. Let $ g:\left(Y, \sigma \right) \longrightarrow (Z, \gamma) $ be an $ \omega _{s} $-open function and let $ V\in \sigma $. Since $ g $ is $ \omega _{s} $-open, then $ g(V)\in \omega _{s}(Z, \gamma)\subseteq SO(Z, \gamma) $.

    The converse of Theorem 3.6 is not true as shown in the next example:

    Example 3.7. Consider $ (\mathbb{R}, \sigma) $ where $ \sigma = \left\{ \emptyset, \mathbb{N}, \mathbb{R} \right\} $. Define $ g:(\mathbb{R}, \sigma)\longrightarrow (\mathbb{R}, \sigma) $ by $ g\left(y\right) = y-1 $. Since $ g\left(\mathbb{N} \right) = \left\{ 0\right\} \cup \mathbb{N} \subseteq \overline{ \mathbb{N} } = \mathbb{R} $, then, $ g\left(\mathbb{N} \right) \in SO\left(\mathbb{R}, \sigma \right) $. Also, $ g\left(\emptyset \right) = \emptyset \in SO\left(\mathbb{R}, \sigma \right) \ $and $ g\left(\mathbb{R} \right) = \mathbb{R} \in SO\left(\mathbb{R}, \sigma \right) $. Therefore, $ g $ is semi-open. Conversely, $ g $ is not $ \omega _{s} $-open since there is $ \mathbb{N} \in \sigma $ such that $ g\left(\mathbb{N} \right) = \left\{ 0\right\} \cup \mathbb{N} \notin \omega _{s}\left(\mathbb{R}, \sigma \right) $.

    Theorem 3.8. If $ g:\left(Y, \sigma \right) \longrightarrow (Z, \gamma) $ is a semi-open function such that $ (Z, \gamma) $ is anti-locally countable, then $ g $ is $ \omega _{s} $-open.

    Proof. Follows from the definitions and Proposition 1.1 (c).

    Definition 3.9. A function $ g:\left(Y, \sigma \right) \longrightarrow (Z, \gamma) $ is called pre-$ \omega _{s} $-open, if for every $ A\in \omega _{s}\left(Y, \sigma \right) $, $ g(A)\in \omega _{s}(Z, \gamma) $.

    The following two examples will show that openness and pre-$ \omega _{s} $ -openness are independent concepts:

    Example 3.10. Let $ X = \mathbb{R} $, $ Y = \left\{ a, b, c\right\} $, $ \tau = \left\{ \emptyset, \left(-\infty, 1\right), X\right\} $, and $ \sigma = \left\{ \emptyset, \left\{ a\right\}, Y\right\} $. Define $ f:(X, \tau)\longrightarrow (Y, \sigma) $ by

    $ f(x)={aif x(,1)bif x=1cif x(1,).
    $

    Since $ f\left(\emptyset \right) = \emptyset \in \sigma $, $ f\left(\left(-\infty, 1\right) \right) = \left\{ a\right\} \in \sigma $, and $ f\left(X\right) = Y\in \sigma $, then $ f $ is open. Since $ (X, \tau) $ is anti-locally countable, then by Proposition 1.3, $ \overline{\left(-\infty, 1\right) } ^{\omega } = \overline{\left(-\infty, 1\right) } = \mathbb{R} $. So, we have $ \left(-\infty, 1\right] \in \omega _{s}\left(X, \tau \right) $ but $ f\left(\left(-\infty, 1\right] \right) = \left\{ a, b\right\} \notin \omega _{s}(Y, \sigma) = \sigma $. This shows that $ f $ is not pre-$ \omega _{s} $-open.

    Example 3.11. Consider $ \left(\mathbb{R}, \tau \right) $ where $ \tau = \left\{ \emptyset, \mathbb{R}, \left[1, \infty \right) \right\} $. Define $ f:\left(\mathbb{R}, \tau \right) \longrightarrow \left(\mathbb{R}, \tau \right) $ by $ f\left(x\right) = x-1 $. Since $ \left(\mathbb{R}, \tau \right) $ is anti-locally countable, then by Proposition 1.1 (c), $ \omega _{s}\left(\mathbb{R}, \tau \right) = SO\left(\mathbb{R}, \tau \right) = \left\{ \emptyset \right\} \cup \left\{ H:\left[1, \infty \right) \subseteq H\right\} $. To see that $ f $ is pre-$ \omega _{s} $-open, let $ H\in \omega _{s}\left(\mathbb{R}, \tau \right) -\left\{ \emptyset \right\} $. Then $ \left[0, \infty \right) = f\left(\left[1, \infty \right) \right) \subseteq f\left(H\right) $, and thus $ \left[1, \infty \right) \subseteq f\left(H\right) $. Hence, $ f\left(H\right) \in \omega _{s}\left(\mathbb{R}, \tau \right) $. Therefore, $ f $ is pre-$ \omega _{s} $-open. Conversely, $ f $ is not open since there is $ \left[1, \infty \right) \in \tau $ such that $ f\left(\left[1, \infty \right) \right) = \left[0, \infty \right) \notin \tau $.

    Theorem 3.12. Let $ \left(Y, \sigma \right) $ and $ (Z, \gamma) $ be locally countable, and let $ g:\left(Y, \sigma \right) \longrightarrow (Z, \gamma) $ be a function. Then the followings are equivalent:

    (a) $ g $ is open.

    (b) $ g $ is pre-$ \omega _{s} $-open.

    Proof. Follows from the definitions and Proposition 1.1 (b).

    The following two examples will show that pre-semi-openness and pre-$ \omega _{s} $-openness are independent concepts:

    Example 3.13. Consider the function $ g $ as in Example 3.7. It is not difficult to see that $ SO\left(\mathbb{R}, \sigma \right) = \left\{ \emptyset \right\} \cup \left\{ H: \mathbb{N} \subseteq H\right\} $ and $ \omega _{s}\left(\mathbb{R}, \sigma \right) = \sigma $. To see that $ g $ is pre-semi-open, let $ H\in SO\left(\mathbb{R}, \sigma \right) -\left\{ \emptyset \right\} $. Then $ \mathbb{N} \subseteq H $ and so $ \mathbb{N} \subseteq \mathbb{N} \cup \left\{ 0\right\} = g\left(\mathbb{N} \right) $. Thus, $ g\left(\mathbb{N} \right) \in SO\left(\mathbb{R}, \sigma \right) $. This shows that $ g $ is pre-semi-open. Conversely, $ g $ is not pre-$ \omega _{s} $-open since there is $ \mathbb{N} \in \omega _{s}\left(\mathbb{R}, \sigma \right) $ such that $ g\left(\mathbb{N} \right) = \left\{ 0\right\} \cup \mathbb{N} \notin \omega _{s}\left(\mathbb{R}, \sigma \right) $.

    Example 3.14. Let $ X = Y = \{1, 2, 3, 4\} $, $ \tau = \{\emptyset, X, \{1, 2\}, \{1\}, \{2\}\} $, and $ \sigma = \{\emptyset, Y, \{2, 3, 4\}, \{1, 2\}, \{1\}, \{2\}\} $. Let $ f:\left(X, \tau \right) \longrightarrow (Y, \sigma) $ be the identity function. It is clear that $ f $ is open. Since $ \left(X, \tau \right) $ and $ (Y, \sigma) $ are locally countable, then by Theorem 3.12, $ f $ is pre-$ \omega _{s} $-open. Conversely, $ f $ is not pre-semi-open because there is $ \left\{ 1, 3\right\} \in \; SO\left(X, \tau \right) $ such that $ f\left(\left\{ 1, 3\right\} \right) = \left\{ 1, 3\right\} \notin SO\left(Y, \sigma \right) $.

    Theorem 3.15. Let $ \left(X, \tau \right) $ and $ (Y, \sigma) $ be anti-locally countable topological spaces and let $ f:\left(X, \tau \right) \longrightarrow (Y, \sigma) $ be a function. Then $ f $ is pre-semi-open if and only if $ f $ is pre-$ \omega _{s} $-open.

    Proof. Follows from definitions and Proposition 1.1 (c).

    Theorem 3.16. Every pre-$ \omega _{s} $-open function is $ \omega _{s} $ -open. Proof. Let $ f:\left(X, \tau \right) \longrightarrow (Y, \sigma) $ be a pre-$ \omega _{s} $-open function and let $ U\in \tau \subseteq \omega _{s}\left(X, \tau \right) $. Since $ f $ is pre-$ \omega _{s} $-open, then $ f(U)\in \omega _{s}(Y, \sigma) $.

    The function $ f $ in Example 3.10 is open but not pre-$ \omega _{s} $-open, and by Theorem 3.3, $ f $ is $ \omega _{s} $-open. Therefore, the converse of the implication in Theorem 3.16 is not true, in general.

    Theorem 3.17. For a function $ g:(Y, \sigma)\longrightarrow (Z, \lambda) $, the followings are equivalent:

    (a) $ g $ is $ \omega _{s} $-open.

    (b) $ g^{-1}\left(\overline{B}^{\omega _{s}}\right) \subseteq \overline{ g^{-1}\left(B\right) } $ for every $ B\subseteq Z $.

    (c) $ Int\left(g^{-1}\left(B\right) \right) \subseteq g^{-1}\left(Int_{\omega _{s}}\left(B\right) \right) $ for every $ B\subseteq Z $.

    Proof. (a) $ \Longrightarrow $ (b): Suppose that $ g $ is $ \omega _{s} $-open and let $ B\subseteq Z $. Let $ y\in g^{-1}\left(\overline{B}^{\omega _{s}}\right) $. To show that $ y\in \overline{ g^{-1}\left(B\right) } $, let $ V\in \sigma $ such that $ y\in V $. Then $ g\left(y\right) \in g\left(V\right) $. Since $ g $ is $ \omega _{s} $-open, then $ g\left(V\right) \in \omega _{s}(Z, \lambda) $. Since $ g\left(y\right) \in g\left(V\right) \cap \overline{B}^{\omega _{s}} $, then $ g\left(V\right) \cap B\neq \emptyset $. Choose $ t\in V $ such that $ g\left(t\right) \in B $. Then $ t\in V\cap g^{-1}\left(B\right) $, and hence $ V\cap g^{-1}\left(B\right) \neq \emptyset $. It follows that $ y\in \overline{ g^{-1}\left(B\right) } $.

    (b) $ \Longrightarrow $ (a): Suppose that $ g^{-1}\left(\overline{B}^{\omega _{s}}\right) \subseteq \overline{g^{-1}\left(B\right) } $ for every $ B\subseteq Z $, and suppose to the contrary that $ g $ is not $ \omega _{s} $-open. Then there exists $ V\in \sigma $ such that $ g\left(V\right) \notin \omega _{s}(Z, \lambda) $ and so, $ Z-g\left(V\right) $ is not $ \omega _{s} $-closed. Thus, there exists $ z\in g\left(V\right) \cap \overline{Z-g\left(V\right) }^{\omega _{s}} $. Choose $ y\in V $ such that $ z = g\left(y\right) $. Then $ y\in g^{-1}\left(\overline{Z-g\left(V\right) } ^{\omega _{s}}\right) $. By assumption, we have

    $ g1(¯Zg(V)ωs)¯g1(Zg(V))=¯Yg1(g(V))¯YV=YV.
    $

    Therefore, $ y\in Y-V $. But $ y\in V $, a contradiction.

    (b) $ \Longrightarrow $ (c): Suppose that $ g^{-1}\left(\overline{B} ^{\omega _{s}}\right) \subseteq \overline{g^{-1}\left(B\right) } $ for every $ B\subseteq Z $. Let $ B\subseteq Z $. Then by (b),

    $ g1(Intωs(B))=g1(Z¯ZBωs)=Yg1(¯ZBωs)Y¯g1(ZB)=Y¯Yg1(B)=Int(g1(B)).
    $

    (c) $ \Longrightarrow $ (b): Suppose that $ Int\left(g^{-1}\left(B\right) \right) \subseteq g^{-1}\left(Int_{\omega _{s}}\left(B\right) \right) $ for every $ B\subseteq Z $. Let $ B\subseteq Z $. Then by (c),

    $ g1(¯Bωs)=g1(ZIntωs(ZB))=Yg1(Intωs(ZB))YInt(g1(ZB))=YInt(Yg1(B))=¯g1(B).
    $

    Theorem 3.18. For a function $ g:(Y, \sigma)\longrightarrow (Z, \lambda) $, the followings are equivalent:

    (a) $ g $ is pre-$ \omega _{s} $-open.

    (b) $ g^{-1}\left(\overline{B}^{\omega _{s}}\right) \subseteq \overline{ g^{-1}\left(B\right) }^{\omega _{s}} $ for every $ B\subseteq Z $.

    (c) $ Int_{\omega _{s}}\left(g^{-1}\left(B\right) \right) \subseteq g^{-1}\left(Int_{\omega _{s}}\left(B\right) \right) $ for every $ B\subseteq Z $.

    Proof. (a) $ \Longrightarrow $ (b): Suppose that $ g $ is pre-$ \omega _{s} $-open and let $ B\subseteq Z $. Let $ y\in g^{-1}\left(\overline{B}^{\omega _{s}}\right) $. To show that $ y\in \overline{ g^{-1}\left(B\right) }^{\omega _{s}} $, let $ C\in \omega _{s}(Y, \sigma) $ such that $ y\in C $. Then $ g\left(y\right) \in g\left(C\right) $. Since $ g $ is pre-$ \omega _{s} $-open, then $ g\left(C\right) \in \omega _{s}(Z, \lambda) $. Since $ g\left(y\right) \in g\left(C\right) \cap \overline{B}^{\omega _{s}} $, then $ g\left(C\right) \cap B\neq \emptyset $. Choose $ t\in C $ such that $ g\left(t\right) \in B $. Then $ t\in C\cap g^{-1}\left(B\right) $ and hence $ C\cap g^{-1}\left(B\right) \neq \emptyset $. It follows that $ y\in \overline{g^{-1}\left(B\right) }^{\omega _{s}} $.

    (b) $ \Longrightarrow $ (a): Suppose that $ g^{-1}\left(\overline{B}^{\omega _{s}}\right) \subseteq \overline{ g^{-1}\left(B\right) }^{\omega _{s}} $ for every $ B\subseteq Z $, and suppose to the contrary that $ g $ is not pre-$ \omega _{s} $-open. Then there is $ C\in \omega _{s}(Y, \sigma) $ such that $ g\left(C\right) \notin \omega _{s}(Z, \lambda) $ and so $ Z-g\left(C\right) $ is not $ \omega _{s} $-closed. And so there exists $ z\in g\left(C\right) \cap \overline{Z-g\left(C\right) }^{\omega _{s}} $. Choose $ y\in C $ such that $ z = g\left(y\right) $. Then $ y\in g^{-1}\left(\overline{Z-g\left(C\right) }^{\omega _{s}}\right) $. By assumption we have

    $ g1(¯Zg(C)ωs)¯g1(Zg(C))ωs=¯Yg1(g(C))ωs¯YCωs=YC.
    $

    Therefore, $ y\in Y-C $ but $ y\in C $, a contradiction.

    (b) $ \Longrightarrow $ (c): Suppose that $ g^{-1}\left(\overline{B} ^{\omega _{s}}\right) \subseteq \overline{g^{-1}\left(B\right) }^{\omega _{s}} $ for every $ B\subseteq Z $. Let $ B\subseteq Z $. Then by (b),

    $ g1(Intωs(B))=g1(Z¯ZBωs)=Yg1(¯ZBωs)Y¯g1(ZB)ωs=Y¯Yg1(B)ωs=Intωs(g1(B)).
    $

    (c) $ \Longrightarrow $ (b): Suppose that $ Int_{\omega _{s}}\left(g^{-1}\left(B\right) \right) \subseteq g^{-1}\left(Int_{\omega _{s}}\left(B\right) \right) $ for every $ B\subseteq Z $. Let $ B\subseteq Z $. Then by (c),

    $ g1(¯Bωs)=g1(ZIntωs(ZB))=Yg1(Intωs(ZB))YIntωs(g1(ZB))=YIntωs(Yg1(B))=¯g1(B)ωs.
    $

    Theorem 3.19. If $ g:(Y, \sigma)\longrightarrow (Z, \lambda) $ is $ \omega _{s} $-continuous such that $ g:(Y, \sigma _{\omega })\longrightarrow (Z, \lambda _{\omega }) $ is open, then $ g:(Y, \sigma)\longrightarrow (Z, \lambda) $ is $ \omega _{s} $-irresolute.

    Proof. Suppose that $ g:(Y, \sigma)\longrightarrow (Z, \lambda) $ is $ \omega _{s} $-continuous with $ g:(Y, \sigma _{\omega })\longrightarrow (Z, \lambda _{\omega }) $ is open. Let $ G\in \omega _{s}(Z, \lambda) $, choose $ W\in \lambda $ such that $ W\subseteq G\subseteq \overline{W}^{\omega } $. Thus, we have $ g^{-1}\left(W\right) \subseteq g^{-1}\left(G\right) \subseteq g^{-1}\left(\overline{W}^{\omega }\right) $. Since $ g:(Y, \sigma)\longrightarrow (Z, \lambda) $ is $ \omega _{s} $ -continuous, then $ g^{-1}\left(W\right) \in \omega _{s}(Y, \sigma) $. Since $ g:(Y, \sigma _{\omega })\longrightarrow (Z, \lambda _{\omega }) $ is open, then $ g^{-1}\left(\overline{W}^{\omega }\right) \subseteq \overline{g^{-1}\left(W\right) }^{\omega } $. Since we have $ g^{-1}\left(W\right) \subseteq g^{-1}\left(G\right) \subseteq \overline{g^{-1}\left(W\right) }^{\omega } $ with $ g^{-1}\left(W\right) \in \omega _{s}(Y, \sigma) $, then by Proposition 1.2 (a), $ g^{-1}\left(G\right) \in \omega _{s}(Y, \sigma) $. This ends the proof.

    Theorem 3.20. If $ g:(Y, \sigma)\longrightarrow (Z, \lambda) $ is an $ \omega _{s} $-open function such that $ g:(Y, \sigma _{\omega })\longrightarrow (Z, \lambda _{\omega }) $ is a continuous function, then $ g:(Y, \sigma)\longrightarrow (Z, \lambda) $ is pre-$ \omega _{s} $-open.

    Proof. Suppose that $ g:(Y, \sigma)\longrightarrow (Z, \lambda) $ is $ \omega _{s} $-open with $ g:(Y, \sigma _{\omega })\longrightarrow (Z, \lambda _{\omega }) $ is continuous. Let $ H\in \omega _{s}(Y, \sigma) $, then we find $ V\in \sigma $ such that $ V\subseteq H\subseteq \overline{V}^{\omega } $. Thus, we have $ g\left(V\right) \subseteq g\left(H\right) \subseteq g\left(\overline{V}^{\omega }\right) $. Since $ g:(Y, \sigma)\longrightarrow (Z, \lambda) $ is $ \omega _{s} $-open, then $ g\left(V\right) \in \omega _{s}(Z, \lambda) $. By continuity of $ g:(Y, \sigma _{\omega })\longrightarrow (Z, \lambda _{\omega }) $, $ g\left(\overline{V}^{\omega }\right) \subseteq \overline{g\left(V\right) }^{\omega } $. Since we have $ g\left(V\right) \subseteq g\left(H\right) \subseteq \overline{g\left(V\right) }^{\omega } $ with $ g\left(V\right) \in \omega _{s}(Z, \lambda) $, then by Proposition 1.2 (a), $ g\left(V\right) \in \omega _{s}(Z, \lambda) $. This ends the proof.

    As defined in [32], a function $ g:(Y, \sigma)\longrightarrow (Z, \lambda) $ is $ \omega $-continuous if for each $ W\in \lambda $, $ g^{-1}(W)\in \sigma _{\omega } $.

    Theorem 3.21. Let $ g:(Y, \sigma)\longrightarrow (Z, \lambda) $ be pre-$ \omega _{s} $-open and $ \omega _{s} $-irresolute such that $ (Z, \lambda) $ is semi-regular and dense in itself, then $ g $ is $ \omega $-continuous.

    Proof. Suppose to the contrary that $ g $ is not $ \omega $ -continuous. Then there is $ W\in \lambda $ such that $ g^{-1}\left(W\right) \notin \sigma _{\omega } $. So, there exists $ y\in g^{-1}\left(W\right) -Int_{\omega }\left(g^{-1}\left(W\right) \right) $. Since $ g\left(y\right) \in W $ and $ (Z, \lambda) $ is semi-regular, then there is a regular open set $ M $ of $ (Z, \lambda) $ such that $ g\left(y\right) \in M\subseteq W $. Since $ Int\left(\overline{M}^{\omega }\right) \subseteq Int\left(\overline{M}\right) = M $, then by Theorem 2.16 of [29], $ M $ is $ \omega _{s} $ -closed. Since $ g $ is $ \omega _{s} $-irresolute, then by Theorem 2.11, $ g^{-1}\left(M\right) $ is $ \omega _{s} $-closed, and so $ Y-g^{-1}\left(M\right) \in \omega _{s}(Y, \sigma) $. Since $ g^{-1}\left(M\right) \subseteq g^{-1}\left(W\right) $, then $ Int_{\omega }\left(g^{-1}\left(M\right) \right) \subseteq Int_{\omega }\left(g^{-1}\left(W\right) \right) $. Since $ y\notin Int_{\omega }\left(g^{-1}\left(W\right) \right) $, then $ y\notin Int_{\omega }\left(g^{-1}\left(M\right) \right) $, and so

    $ yYIntω(g1(M))=¯Yg1(M)ω.
    $

    Thus, by Proposition 1.2 (a), we have $ \left(Y-g^{-1}\left(M\right) \right) \cup \left\{ y\right\} \in \omega _{s}(Y, \sigma) $ with $ y\notin Y-g^{-1}\left(M\right) $. Put $ S = g\left(\left(Y-g^{-1}\left(M\right) \right) \cup \left\{ y\right\} \right) = g\left(Y-g^{-1}\left(M\right) \right))\cup \left\{ g\left(y\right) \right\} $. Since $ g $ is pre-$ \omega _{s} $-open, then $ S\in \omega _{s}(Z, \lambda) $. So by Proposition 1.2 (b), we have $ S\cap M\in \omega _{s}(Z, \lambda) $. Since $ g\left(Y-g^{-1}\left(M\right) \right))\subseteq Z-M $, then we have

    $ g(y)SM((ZM){g(y)})M={g(y)}
    $

    and thus $ S\cap M = \left\{ g\left(y\right) \right\} $. Therefore, $ \left\{ g\left(y\right) \right\} \in \omega _{s}(Z, \lambda) $. Thus, there exists $ O\in \lambda $ such that $ O\subseteq \left\{ g\left(y\right) \right\} \subseteq \overline{O}^{\omega } $, and hence $ O = \left\{ g\left(y\right) \right\} $. This implies that $ \left\{ g\left(y\right) \right\} \in \lambda $. But by assumption $ (Z, \lambda) $ is dense in itself, a contradiction.

    The condition '$ (Z, \lambda) $ is dense in itself' in Theorem 3.21 cannot be dropped as our next example shows:

    Example 3.22. Take $ f $ as in Example 2.7. Then $ f $ is $ \omega _{s} $ -irresolute and pre-$ \omega _{s} $-open. Also, $ (\left\{ a, b\right\}, \tau _{disc}) $ is semi-regular. On the other hand, since $ \left\{ b\right\} \in \tau _{disc} $ but $ f^{-1}\left(\left\{ b\right\} \right) = \left[0, \infty \right) \notin \left(\tau _{u}\right) _{\omega } $, then $ f $ is not $ \omega $ -continuous.

    Theorem 3.23. Let $ g:(Y, \sigma)\longrightarrow (Z, \lambda) $ be injective, pre-$ \omega _{s} $-open, and $ \omega _{s} $ -irresolute such that $ (Z, \lambda) $ is semi-regular, then $ g $ is $ \omega $ -continuous.

    Proof. Suppose to the contrary that $ g $ is not $ \omega $ -continuous. Then there is $ W\in \lambda $ such that $ g^{-1}\left(W\right) \notin \sigma _{\omega } $. So, there exists $ y\in g^{-1}\left(W\right) -Int_{\omega }\left(g^{-1}\left(W\right) \right) $. Since $ g\left(y\right) \in W $ and $ (Z, \lambda) $ is semi-regular, then there is a regular open set $ M $ of $ (Z, \lambda) $ such that $ g\left(y\right) \in M\subseteq W $. Since $ Int\left(\overline{M}^{\omega }\right) \subseteq Int\left(\overline{M}\right) = M $, then by Theorem 2.16 of [29], $ M $ is $ \omega _{s} $ -closed. Since $ g $ is $ \omega _{s} $-irresolute, then by Theorem 2.11, $ g^{-1}\left(M\right) $ is $ \omega _{s} $-closed, and so $ Y-g^{-1}\left(M\right) \in \omega _{s}\left(Y, \sigma \right) $. Since $ g^{-1}\left(M\right) \subseteq g^{-1}\left(W\right) $, then $ Int_{\omega }\left(g^{-1}\left(M\right) \right) \subseteq Int_{\omega }\left(g^{-1}\left(W\right) \right) $. Since $ y\notin Int_{\omega }\left(g^{-1}\left(W\right) \right) $, then $ y\notin Int_{\omega }\left(g^{-1}\left(M\right) \right) $, and so

    $ yYIntω(g1(M))=¯Yg1(M)ω.
    $

    Thus, by Proposition 1.2 (a), we have $ \left(Y-g^{-1}\left(M\right) \right) \cup \left\{ y\right\} \in \omega _{s}\left(Y, \sigma \right) $ with $ y\notin Y-g^{-1}\left(M\right) $. Put $ S = g\left(\left(Y-g^{-1}\left(M\right) \right) \cup \left\{ y\right\} \right) = g\left(Y-g^{-1}\left(M\right) \right))\cup \left\{ g\left(y\right) \right\} $. Since $ g $ is pre- $ \omega _{s} $-open, then $ S\in \omega _{s}(Z, \lambda) $, and by Proposition 1.2 (b) we have $ S\cap M\in \omega _{s}(Z, \lambda) $. Since $ g\left(Y-g^{-1}\left(M\right) \right))\subseteq Z-M $, then we have

    $ g(y)SM((ZM){g(y)})M={g(y)}
    $

    and thus $ S\cap M = \left\{ g\left(y\right) \right\} $. Therefore, $ \left\{ g\left(y\right) \right\} \in \omega _{s}(Z, \lambda) $. Since $ g $ is $ \omega _{s} $-irresolute and injective, then $ g^{-1}\left(\left\{ g\left(y\right) \right\} \right) = \left\{ y\right\} \in \omega _{s}\left(Y, \sigma \right) $. So, there exists $ V\in \sigma $ such that $ V\subseteq \left\{ y\right\} \subseteq \overline{V}^{\omega } $, and hence $ \left\{ y\right\} \in \sigma $. Since $ y\in g^{-1}\left(W\right) $ and $ \left\{ y\right\} \in \sigma \subseteq \sigma _{\omega } $, then $ y\in Int_{\omega }\left(g^{-1}\left(W\right) \right) $, a contradiction.

    Example 3.23 shows that the condition 'injective' in Theorem 3.23 cannot be dropped.

    Definition 3.24. A function $ g:(Y, \sigma)\longrightarrow (Z, \lambda) $ is called$ \ \omega _{s} $-closed if for each $ \omega _{s} $ -closed set $ C $ of $ (Y, \sigma) $, $ g(C) $ is $ \omega _{s} $-closed set in $ (Z, \lambda) $.

    As defined in [33] a topological space $ (X, \tau) $ is called $ \omega $ -regular if for each closed set $ F $ in $ (X, \tau) $ and $ x\in X-F $, there exist $ U \; \in \; \tau $ and $ V\in \; \tau _{\omega } $ such that $ x\in U $, $ F\subseteq V $ and $ U\cap V = \emptyset $. As defined in [31], a function $ g:(Y, \sigma)\longrightarrow (Z, \lambda) $ is $ \omega $-open if for each $ V\in \sigma $, $ g(V)\in \lambda _{\omega } $.

    Theorem 3.25. If $ g:(Y, \sigma)\longrightarrow (Z, \lambda) $ is $ \omega _{s} $-closed, pre-$ \omega _{s} $-open, and $ \omega _{s} $ -irresolute such that $ (Y, \sigma) $ is $ \omega $-regular, then $ g $ is $ \omega $-open.

    Proof. Suppose to the contrary that there exists $ V\in \sigma $ such that $ g\left(V\right) \notin \lambda _{\omega } $. Then we find $ y\in V $ such that $ g\left(y\right) \in g\left(V\right) -Int_{\omega }\left(g\left(V\right) \right) $. By $ \omega $-regularity of $ (Y, \sigma) $, we find $ M\in \sigma $ such that $ y\in M\subseteq \overline{M}^{\omega }\subseteq V $. Since $ \overline{M}^{\omega } $ is $ \omega _{s} $-closed and $ g $ is $ \omega _{s} $-closed, then $ Z-g\left(\overline{M}^{\omega }\right) \in \omega _{s}(Z, \lambda) $ with $ g\left(y\right) \notin Z-g\left(\overline{M} ^{\omega }\right) $. Since $ g\left(y\right) \notin Int_{\omega }\left(g\left(V\right) \right) $, then $ g\left(y\right) \notin Int_{\omega }\left(g\left(\overline{M}^{\omega }\right) \right) $. Thus,

    $ g(y)ZIntω(g(¯Mω))=¯Zg(¯Mω)ω.
    $

    So by Proposition 1.2 (a), $ \left(Z-g\left(\overline{M}^{\omega }\right) \right) \cup \left\{ g\left(y\right) \right\} \in \omega _{s}(Z, \lambda) $. Set $ B = g^{-1}\left(\left(Z-g\left(\overline{M}^{\omega }\right) \right) \cup \left\{ g\left(y\right) \right\} \right) $. Since $ g $ is $ \omega _{s} $ -irresolute, then $ B\in \omega _{s}(Y, \sigma) $ and by Proposition 1.2 (b), $ M\cap B\in \omega _{s}(Y, \sigma) $. Since $ g $ is pre-$ \omega _{s} $-open, then $ g\left(M\cap B\right) \in \omega _{s}(Z, \lambda) $. Since

    $ g(y)g(MB)g(M)g(B)g(M)((Zg(¯Mω)){g(y)})={g(y)}
    $

    then $ \left\{ g\left(y\right) \right\} \in \omega _{s}(Z, \lambda) $. Thus, there is $ K\in \lambda $ such that $ K\subseteq \left\{ g\left(y\right) \right\} \subseteq \overline{K}^{\omega } $. Hence, $ \left\{ g\left(y\right) \right\} \in \lambda $. Since $ g\left(y\right) \in g\left(V\right) $, then $ g\left(y\right) \in Int\left(g\left(V\right) \right) \subseteq Int_{\omega }\left(g\left(V\right) \right) $, a contradiction.

    Let $ (Y, \sigma) $ be a topological space and let $ B $ be a subset of $ Y $. Then $ B $ is called clopen (resp. semi-clopen, $ \omega _{s} $-clopen) in $ (Y, \sigma) $ if $ B $ both open and closed (resp. semi-open and semi-closed, $ \omega _{s} $-open and $ \omega _{s} $-closed) in $ (Y, \sigma) $. Throughout this section, the family of all clopen (resp., semi-clopen, $ \omega _{s} $ -clopen) subsets of the topological space $ (Y, \sigma) $ will be denoted by $ CO(Y, \sigma) $ (resp. $ SCO(Y, \sigma) $, $ \omega _{s}CO(Y, \sigma) $).

    Definition 4.1. A function $ g:(Y, \sigma)\longrightarrow (Z, \lambda) $ is called slightly continuous [34] (resp. slightly semi-continuous [15], slightly $ \omega _{s} $-continuous), if for every $ y \; \in \; Y $ and every $ W\in CO(Z, \lambda) $ with $ g\left(y\right) \in W $, there exists $ V\in \sigma $ (resp. $ V\in SO(Y, \sigma) $, $ V\in \omega _{s}(Y, \sigma) $) such that $ y\in V $ and $ g(V) \; \subseteq \; W $.

    As an example of a slightly continuous function that is not continuous, take the function $ g:(\mathbb{R}, \tau _{u})\rightarrow (\mathbb{R}, \tau _{u}) $ defined by $ g(y) = [y] $, where $ [y] $ is the greatest integer of $ y $.

    Theorem 4.2. For a function $ g:(Y, \sigma)\longrightarrow (Z, \lambda) $, the followings are equivalent:

    (a) $ g $ is slightly $ \omega _{s} $-continuous.

    (b) For all $ W\in CO(Z, \lambda) $, $ g^{-1}\left(W\right) \in \omega _{s}(Y, \sigma) $.

    (c) For all $ W\in CO(Z, \lambda) $, $ g^{-1}\left(W\right) \in \omega _{s}CO(Y, \sigma) $.

    Proof. (a) $ \Longrightarrow $ (b): Let $ W\in CO(Z, \lambda) $. Then for each $ y\in g \; ^{-1}(W) $, $ g(y)\in W $ and by (a), there exists $ V_{y}\in \omega _{s}(Y, \sigma) $ such that $ y \; \in \; V_{y} $ and $ g(V_{y}) \; \subseteq \; W $. Thus,

    $ g1(W)={Vy:yg1(W)}.
    $

    Therefore, $ g^{-1}(W) $ is a union of $ \omega _{s} $-open sets, and hence $ g^{-1}(W) $ is $ \omega _{s} $-open.

    (b) $ \Longrightarrow $ (c): Let $ W\in CO(Z, \lambda) $. Then $ Z-W \; \in CO(Z, \lambda) $. Thus, by (b), $ g^{-1}(W)\in \omega _{s}(Y, \sigma) $ and $ g^{-1}(Z \; -W) = Y-g^{-1}(W)\in \omega _{s}(Y, \sigma) $. Therefore, $ g^{-1}(W)\in \omega _{s}CO(Y, \sigma) $.

    (c) $ \Longrightarrow $ (a): Let $ y\in Y $ and $ W\in CO(Z, \lambda) $ with $ g(y) \; \in \; W $. By (c), $ g^{-1}(W)\in \omega _{s}CO(Y, \sigma)\subseteq \omega _{s}(Y, \sigma) $. Put $ V \; = \; g^{-1}(W) $. Then $ V\in \omega _{s}(Y, \sigma) $, $ y\in V $, and $ g(V) = g(g \; ^{-1}(W))\subseteq W $. This shows that $ g $ is slightly $ \omega _{s} $-continuous.

    Theorem 4.3. Every slightly continuous function is slightly $ \omega _{s} $-continuous.

    Proof. Assume that $ g:(Y, \sigma)\longrightarrow (Z, \lambda) $ is slightly continuous. Let $ W\in CO(Z, \lambda) $. Since $ g $ is slightly continuous, then $ g^{-1}(W)\in \sigma $. So by Proposition 1.1 (a), $ g^{-1}(W)\in \omega _{s}(Y, \sigma) $. Therefore, by Theorem 4.2, it follows that $ g $ is slightly $ \omega _{s} $-continuous.

    Our next example shows that the converse of Theorem 4.3 is not true, in general:

    Example 4.4. Let $ X = Y = \mathbb{R} $, $ \tau = \left\{ \emptyset, \mathbb{R}, \mathbb{N}, \mathbb{Q} ^{c}, \mathbb{N} \cup \mathbb{Q} ^{c}\right\} $, and $ \sigma \; = \{\emptyset, \mathbb{R}, \mathbb{N}, \mathbb{R} - \mathbb{N} \} $. Define $ f\ :(\mathbb{R}, \tau) \; \rightarrow \; (\mathbb{R}, \sigma) $ by $ f(x) = x $. Note that $ CO(\mathbb{R}, \sigma) = \sigma $. Since $ f^{-1}(\mathbb{N}) = \mathbb{N} \in \tau \subseteq \omega _{s}(X, \tau) $ and $ f^{-1}(\mathbb{R} - \mathbb{N}) = \mathbb{R} - \mathbb{N} \in \; \omega _{s}(X, \tau) $, then $ f $ is$ \ $slightly $ \omega _{s} $ -continuous. On the other hand, since $ \mathbb{R} - \mathbb{N} \in CO(\mathbb{R}, \sigma) $ but $ f^{-1}(\mathbb{R} - \mathbb{N}) \; = \; \mathbb{R} \; - \; \mathbb{N} \; \notin \; \tau $, then $ f $ is not slightly continuous.

    Theorem 4.5. Every slightly $ \omega _{s} $-continuous function is slightly semi-continuous.

    Proof. Assume that $ g:(Y, \sigma)\longrightarrow (Z, \lambda) $ is slightly $ \omega _{s} $-continuous. Let $ W\in CO(Z, \lambda) $. Since $ g $ is slightly $ \omega _{s} $-continuous, then $ g^{-1}(W)\in \omega _{s}(Y, \sigma) $. So by Proposition 1.1 (a), $ g^{-1}(W)\in SO(Y, \sigma) $. Therefore, $ g $ is slightly semi-continuous.

    The converse of Theorem 4.5 is not true in general as the following example shows:

    Example 4.6. Let $ X = Y = \mathbb{R} $, $ \ \tau = \left\{ \emptyset, \mathbb{R}, \mathbb{N}, \mathbb{Q} ^{c}, \mathbb{N} \cup \mathbb{Q} ^{c}\right\} $, and $ \sigma \; = \{\emptyset, \mathbb{R}, \mathbb{Q}, \mathbb{R} - \mathbb{Q} \} $. Define $ f\ :(\mathbb{R}, \tau) \; \rightarrow \; (\mathbb{R}, \sigma) $ by $ f(x) = x $. Note that $ CO(\mathbb{R}, \sigma) = \; \sigma $. Since $ f^{-1}(\mathbb{Q}) = \mathbb{Q} \in SO(X, \tau) $ and $ f^{-1}(\mathbb{R} - \mathbb{Q}) = \mathbb{R} - \mathbb{Q} \in \; \tau \subseteq SO(X, \tau) $, then $ f $ is$ \ $slightly semi-continuous. On the other hand, since $ \mathbb{Q} \in CO(\mathbb{R}, \sigma) $ but $ f^{-1}(\mathbb{Q}) \; = \; \mathbb{Q} \; \notin \; \omega _{s}(\mathbb{R}, \tau) $, then $ f $ is not slightly $ \omega _{s} $-continuous.

    Theorem 4.7.(a) If $ g:(Y, \sigma)\longrightarrow (Z, \lambda) $ is slightly $ \omega _{s} $-continuous such that $ (Y, \sigma) $ is locally countable, then $ g $ is continuous.

    (b) If $ g:(Y, \sigma)\longrightarrow (Z, \lambda) $ is slightly semi-continuous with $ (Y, \sigma) $ is anti-locally countable, then $ g $ is $ \omega _{s} $-continuous.

    Proof. (a) Follows from the definitions and Proposition 1.1 (c).

    (b) Follows from the definitions and Proposition 1.1 (b).

    Theorem 4.8. Let $ g:(Y, \sigma)\longrightarrow (Z, \lambda) $ be a function and let $ \gamma $ be the product topology of $ (Y, \sigma) $ and $ (Z, \lambda) $. Let $ h:(Y, \sigma)\longrightarrow \left(Y\times Z, \gamma \right) $, where $ h(y) = (y, g(y)) $ be the graph of $ g $. Then $ g $ is slightly $ \omega _{s} $-continuous if and only if $ h $ is slightly $ \omega _{s} $-continuous.

    Proof. Let $ y \; \in \; Y $ and let $ M\in CO\left(Y\times Z, \gamma \right) $ such that $ h(y) = (y, g(y))\in M $. Then $ M\cap \left(\{y\}\times Z\right) $ is a clopen set in $ \{y\}\times Z $ which contains $ h(y) = (y, g(y)) $. Since $ \{y\}\times Z $ is homeomorphic to $ Z $, then $ \{z\in Z:(y, z)\in M\}\in CO(Z, \lambda) $. Since $ g $ is slightly $ \omega _{s} $-continuous, then $ \cup \{g^{-1}(z):(y, z)\in M\}\in \omega _{s}(Y, \sigma) $. Moreover, $ y\in \cup \; \{g^{-1}(z):(y, z)\in M\}\subseteq h^{-1}(M) $. Hence, $ h^{-1}(M)\in \omega _{s}(Y, \sigma) $. It follows that $ h $ is slightly $ \omega _{s} $-continuous.

    Conversely, let $ H\in CO(Z, \lambda) $. Then $ Y\times H\in CO\left(Y\times Z, \gamma \right) $. By slight $ \omega _{s} $-continuity of $ h $, $ h^{-1}(Y\times H)\in \omega _{s}CO(Y, \sigma) $. Also, $ h^{-1}(Y\times H) \; = g^{-1}(H) $. It follows that $ g $ is slightly $ \omega _{s} $-continuous.

    Theorem 4.9. If $ g:(Y, \sigma)\longrightarrow (Z, \lambda) $ is slightly $ \omega _{s} $-continuous and $ h \; :(Z, \lambda)\longrightarrow \left(W, \delta \right) $ is slightly continuous, then $ h\circ g:(Y, \sigma)\longrightarrow \left(W, \delta \right) $ is slightly $ \omega _{s} $ -continuous.

    Proof. Let $ M\in CO\left(W, \delta \right) $. By slight continuity of $ h $, $ g^{-1}(M)\in CO(Z, \lambda) $. By slight $ \omega _{s} $-continuity of $ g $, $ g^{-1}(h^{-1}(M)) = (h\circ g)^{-1}(M)\in \omega _{s}CO(Y, \sigma) $. Hence, $ h\circ g $ is slightly $ \omega _{s} $-continuous.

    As defined a topological space $ \left(Y, \sigma \right) $ is called semi-connected if $ SCO\left(Y, \sigma \right) = \left\{ \emptyset, Y\right\} $.

    Definition 4.10. A topological space $ \left(Y, \sigma \right) $ is called $ \omega _{s} $-connected if $ \omega _{s}CO\left(Y, \sigma \right) = \left\{ \emptyset, Y\right\} $.

    Theorem 4.11. Every $ \omega _{s} $-connected topological space is connected.

    Proof. Let $ \left(Y, \sigma \right) $ be $ \omega _{s} $-connected. Then $ \omega _{s}CO\left(Y, \sigma \right) = \left\{ \emptyset, Y\right\} $. Thus, by Proposition 1.1 (a), we have $ \left\{ \emptyset, Y\right\} \subseteq CO\left(Y, \sigma \right) \subseteq \omega _{s}CO(Y, \sigma) = \left\{ \emptyset, Y\right\} $. Hence, $ CO(Y, \sigma) = \left\{ \emptyset, Y\right\} $. Therefore, $ \left(Y, \sigma \right) $ is connected.

    The following example will show that the converse of Theorem 4.11 is not true, in general:

    Example 4.12. Consider the topological space $ \left(\mathbb{R}, \tau _{u}\right) $. To see that $ \left(\mathbb{R}, \tau _{u}\right) $ is not $ \omega _{s} $-connected, let $ M = \left(-\infty, 0\right) $, then $ M\in \tau _{u}\subseteq \omega _{s}\left(\mathbb{R}, \tau _{u}\right) $. Since $ \left(0, \infty \right) \in \tau _{u} $ and $ \overline{\left(0, \infty \right) }^{\omega } = \overline{\left(0, \infty \right) } = \left[0, \infty \right) = \mathbb{R} -M $, then $ \mathbb{R} -M\in \omega _{s}\left(\mathbb{R}, \tau _{u}\right) $. Therefore, $ M\in \omega _{s}CO\left(\mathbb{R}, \tau _{u}\right) -\left\{ \emptyset, \mathbb{R} \right\} $, and hence $ \left(\mathbb{R}, \tau _{u}\right) $ is not $ \omega _{s} $-connected. On the other hand, $ \left(\mathbb{R}, \tau _{u}\right) $ is connected.

    Theorem 4.13. Every connected locally countable topological space is $ \omega _{s} $-connected.

    Proof. Follows from the definitions and Proposition 1.1 (b).

    Theorem 4.14. Every semi-connected topological space is $ \omega _{s} $-connected.

    Proof. Let $ \left(Y, \sigma \right) $ be semi-connected. Then $ SCO\left(Y, \sigma \right) = \left\{ \emptyset, Y\right\} $. Thus, by Proposition 1.1 (a), we have $ \left\{ \emptyset, Y\right\} \subseteq \omega _{s}CO\left(Y, \sigma \right) \subseteq SCO(Y, \sigma) = \left\{ \emptyset, Y\right\} $. Hence, $ \omega _{s}CO(Y, \sigma) = \left\{ \emptyset, Y\right\} $. Therefore, $ \left(Y, \sigma \right) $ is $ \omega _{s} $-connected.

    Question 4.15. Is it true that $ \omega _{s} $-connected topological spaces are semi-connected?

    The following theorem answers Question 4.15 partially:

    Theorem 4.16. Every anti-locally countable $ \omega _{s} $-connected topological space is semi-connected.

    Proof. Follows from the definitions and Proposition 1.1 (c).

    Theorem 4.17. A slightly $ \omega _{s} $-continuous image of an $ \omega _{s} $-connected space is connected.

    Proof. Let $ g:(Y, \sigma)\longrightarrow (Z, \lambda) $ be surjective and slightly $ \omega _{s} $-continuous, where $ (Y, \sigma) $ is $ \omega _{s} $-connected. Suppose that $ (Z, \lambda) $ is not connected. Then there exists $ M\in CO(Z, \lambda)-\left\{ \emptyset, Z\right\} $. By Theorem 4.2, $ g^{-1}(M)\in \omega _{s}CO(Y, \sigma) $. Since $ \emptyset \neq M\neq Z $ and $ g $ is surjective, then $ \emptyset \neq g^{-1}\left(M\right) \neq Z $. Therefore, $ g^{-1}\left(M\right) \in \omega _{s}CO(Y, \sigma)-\left\{ \emptyset, Y\right\} $ which contradicts the assumption that $ (Y, \sigma) $ is $ \omega _{s} $-connected.

    Definition 5.1. A topological space $ \left(Y, \sigma \right) $ is called $ \omega _{s} $-compact (resp. semi-compact [36]) if for any cover $ \mathcal{A} $ of $ Y $ with $ \mathcal{A}\subseteq \omega _{s}\left(Y, \sigma \right) $ (resp. $ \mathcal{A}\subseteq SO\left(Y, \sigma \right) $), there is a finite subfamily $ \mathcal{B}\subseteq \mathcal{A} $ such that $ \mathcal{ B} $ is also a cover of $ Y $.

    Theorem 5.2. Every $ \omega _{s} $-compact topological space is compact.

    Proof. Let $ \left(Y, \sigma \right) $ be $ \omega _{s} $ -compact and let $ \mathcal{A} $ be a cover of $ Y $ with $ \mathcal{A}\subseteq \sigma $. Then by Proposition 1.1 (a), $ \mathcal{A}\subseteq \omega _{s}\left(Y, \sigma \right) $. Since $ \left(Y, \sigma \right) $ is $ \omega _{s} $-compact, then there exists a finite subfamily $ \mathcal{B}\subseteq \mathcal{A} $ such that $ \mathcal{B} $ is also a cover of $ Y $. This shows that $ \left(Y, \sigma \right) $ is compact.

    The following example will show that the converse of Theorem 5.2 is not true, in general:

    Example 5.3. Consider $ (\left[0, \infty \right), \sigma) $, where $ \sigma = \left\{ \emptyset, \left[0, \infty \right) \right\} \cup \left\{ \left(a, \infty \right) :a\geq 0\right\} $. To see that $ (\left[0, \infty \right), \sigma) $ is compact, let $ \mathcal{A} $ be a cover of $ \left[0, \infty \right) $ with $ \mathcal{A}\subseteq \sigma $. Then $ \left[0, \infty \right) \in \mathcal{A} $. Choose $ \mathcal{B} = \left\{ \left[0, \infty \right) \right\} $. Then $ \mathcal{B} $ is a finite subfamily of $ \mathcal{A} $ such that $ \mathcal{B} $ is also a cover of $ \left[0, \infty \right) $. This shows that $ (\left[0, \infty \right), \sigma) $ is compact. Let $ \mathcal{A} = \left\{ \left(1, \infty \right) \cup \left\{ x\right\} :x\in \left[0, 1\right] \right\} $. Then $ \mathcal{A} $ is a cover of $ \left[0, \infty \right) $. Since $ \left(1, \infty \right) \in \sigma \subseteq \omega _{s}\left(\left[0, \infty \right), \sigma \right) $ and $ \overline{ \left(1, \infty \right) }^{\omega } = \overline{\left(1, \infty \right) } = \; \left[0, \infty \right) $, then by Proposition 1.2 (a), we have $ \mathcal{A} \subseteq \omega _{s}\left(\left[0, \infty \right), \sigma \right) $. On the other hand, if $ \mathcal{B} $ is a finite subfamily of $ \mathcal{A} $, then $ \mathcal{B} $ is not a cover of $ \left[0, \infty \right) $. This shows that $ (\left[0, \infty \right), \sigma) $ is not $ \omega _{s} $-compact.

    Theorem 5.4. Let $ \left(Y, \sigma \right) $ be a locally countable topological space. Then $ \left(Y, \sigma \right) $ is $ \omega _{s} $ -compact if and only if $ \left(Y, \sigma \right) $ is compact.

    Proof. Necessity. Follows from Theorem 5.2.

    Sufficiency. Suppose that $ \left(Y, \sigma \right) $ is compact and let $ \mathcal{A} $ be a cover of $ Y $ such that $ \mathcal{A} \subseteq \omega _{s}\left(Y, \sigma \right) $. Since $ \left(Y, \sigma \right) $ is locally countable, then by Proposition 1.1 (b), $ \mathcal{A} \subseteq \sigma $. Since $ \left(Y, \sigma \right) $ is compact, then there exists a finite subfamily $ \mathcal{B}\subseteq \mathcal{A} $ such that $ \mathcal{B} $ is also a cover of $ Y $. This shows that $ \left(Y, \sigma \right) $ is $ \omega _{s} $-compact.

    Theorem 5.5. Every semi-compact topological space is $ \omega _{s} $ -compact.

    Proof. Let $ \left(Y, \sigma \right) $ be semi-compact and let $ \mathcal{A} $ be a cover of $ Y $ such that $ \mathcal{A}\subseteq \omega _{s}\left(Y, \sigma \right) $. Then by Proposition 1.1 (a), $ \mathcal{A} \subseteq SO\left(Y, \sigma \right) $. Since $ \left(Y, \sigma \right) $ is semi-compact, then there exists a finite subfamily $ \mathcal{B}\subseteq \mathcal{A} $ such that $ \mathcal{B} $ is also a cover of $ Y $. This shows that $ \left(Y, \sigma \right) $ is $ \omega _{s} $-compact.

    The following example will show that the converse of Theorem 5.5 is not true, in general:

    Example 5.6. Consider $ (\mathbb{N}, \sigma) $, where $ \sigma = \left\{ \emptyset, \mathbb{N}, \left\{ 1\right\}, \left\{ 2\right\}, \left\{ 1, 2\right\} \right\} $. Since $ \left\{ 1\right\} $, $ \left\{ 2\right\} $, and $ \left\{ 1, 2\right\} $ are countable sets, then $ \overline{\left\{ 1\right\} }^{\omega } = \left\{ 1\right\} $, $ \overline{\left\{ 2\right\} }^{\omega } $, and $ \overline{ \left\{ 1, 2\right\} }^{\omega } = \; \left\{ 1, 2\right\} $. Thus, $ \sigma = \omega _{s}\left(Y, \sigma \right) $. This shows that $ (\mathbb{N}, \sigma) $ is $ \omega _{s} $-compact. Let $ \mathcal{A} = \left\{ \left\{ 2\right\} \right\} \cup \left\{ \left\{ 1, x\right\} :x\in \mathbb{N} -\left\{ 1, 2\right\} \right\} $. Then $ \mathcal{A} $ is a cover of $ \mathbb{N} $. Since $ \overline{\left\{ 1\right\} } = \mathbb{N} -\left\{ 2\right\} $, then $ \mathcal{A}\subseteq SO\left(Y, \sigma \right) $. If $ \mathcal{B}\ $is a finite subfamily of $ \mathcal{A} $, then $ \bigcup \mathcal{B} $ is a finite subset of $ \mathbb{N} $. This shows that $ (\mathbb{N}, \sigma) $ is not semi-compact.

    Theorem 5.7. Let $ \left(Y, \sigma \right) $ be an anti-locally countable topological space. Then $ \left(Y, \sigma \right) $ is semi-compact if and only if $ \left(Y, \sigma \right) $ is $ \omega _{s} $ -compact.

    Proof. Necessity. Follows from Theorem 2.5.

    Sufficiency. Suppose that $ \left(Y, \sigma \right) $ is $ \omega _{s} $-compact and let $ \mathcal{A} $ be a cover of $ Y $ such that $ \mathcal{A}\subseteq SO\left(Y, \sigma \right) $. Since $ \left(Y, \sigma \right) $ is anti-locally countable, then by Proposition 1.1 (c), $ \mathcal{A }\subseteq \omega _{s}\left(Y, \sigma \right) $. Since $ \left(Y, \sigma \right) $ is $ \omega _{s} $-compact, then there is a finite subfamily $ \mathcal{B}\subseteq \mathcal{A} $ such that $ \mathcal{B} $ is also a cover of $ Y $. This shows that $ \left(Y, \sigma \right) $ is semi-compact.

    Theorem 5.8. A topological space $ \left(Y, \sigma \right) $ is $ \omega _{s} $-compact if and only if every family of $ \omega _{s} $ -closed sets which has the finite intersection property must have non-empty intersection.

    Proof. Necessity. Suppose that $ \left(Y, \sigma \right) $ is $ \omega _{s} $-compact, and suppose to the contrary that there exists a family $ \mathcal{H} $ of $ \omega _{s} $-closed such that $ \mathcal{H} $ has the finite intersection property and $ \bigcap \mathcal{H} = \emptyset $. Let $ \mathcal{A} = \left\{ Y-H:H\in \mathcal{H}\right\} $. Then $ \mathcal{A} $ is a cover of $ Y $ and $ \mathcal{A}\subseteq \omega _{s}\left(Y, \sigma \right) $. Since $ \left(Y, \sigma \right) $ is $ \omega _{s} $-compact, then there is a finite subfamily $ \mathcal{A}_{1}\subseteq \mathcal{A} $ such that $ \mathcal{A }_{1} $ is also a cover of $ Y $. Let $ \mathcal{H}_{1}\mathcal{ = }\left\{ Y-A:A\in \mathcal{A}_{1}\right\} $. Then $ \mathcal{H}_{1} $ is a finite subcollection of $ \mathcal{H} $ such that

    $ H1=AA1(YA)=YAA1A=YY=.
    $

    This contradicts the assumption that $ \mathcal{H} $ has the finite intersection property.

    Sufficiency. Suppose that every family of $ \omega _{s} $-closed sets which has the finite intersection property must have non-empty intersection, and suppose to the contrary that $ \left(Y, \sigma \right) $ is not $ \omega _{s} $-compact. Then there is a cover $ \mathcal{A} $ of $ Y $ such that $ \mathcal{A}\subseteq \omega _{s}\left(Y, \sigma \right) $ and any finite subcollection of $ \mathcal{A} $ is not a cover of $ Y $. Let $ \mathcal{H} = \left\{ Y-A:A\in \mathcal{A}\right\} $. Then $ \mathcal{H} $ is a family of $ \omega _{s} $-closed sets and $ \mathcal{H} $ has the finite intersection property. So, by assumption $ \bigcap \mathcal{H} \; \neq \; \emptyset $, and thus $ Y-\bigcap \mathcal{H}\neq Y $. But

    $ YH=AAAY
    $

    a contradiction.

    Definition 5.9. Let $ \left(Y, \sigma \right) $ be a topological space and let $ \left(x_{d}\right) _{d\in D} $ be a net in $ \left(Y, \sigma \right) $. A point $ y\in Y $ is called an $ \omega _{s} $ -cluster point of $ \left(y_{d}\right) _{d\in D} $ in $ \left(Y, \sigma \right) $ if for every $ V\in \omega _{s}\left(Y, \sigma \right) $ with $ y\in V $ and every $ d\in D $, there is $ d_{0}\in D $ such that $ d\leq d_{0} $ and $ y_{d_{0}}\in V $.

    Theorem 5.10. A topological space $ \left(Y, \sigma \right) $ is $ \omega _{s} $-compact if and only if every net in $ \left(Y, \sigma \right) $ has an $ \omega _{s} $-cluster point.

    Proof. Necessity. Suppose that $ \left(Y, \sigma \right) $ is $ \omega _{s} $-compact and let $ \left(y_{d}\right) _{d\in D} $ be a net in $ \left(Y, \sigma \right) $. For each $ d\in D $, let $ T_{d} = \left\{ y_{d^{\prime }}:d^{\prime }\in D\text{ and }d\leq d^{\prime }\right\} $. Let $ \mathcal{A} = \left\{ \overline{T_{d}}^{\omega _{s}}:d\in D\right\} $. Then $ \mathcal{A} $ is a family of $ \omega _{s} $-closed sets.

    Claim 1. $ \mathcal{A} $ has the finite intersection property.

    Proof of Claim 1. Let $ d_{1}, d_{2}, ..., d_{n}\in D $. Choose $ d_{0}\in D $ such that $ d_{i}\leq d_{0} $ for all $ i = 1, 2, ..., n $. Then $ y_{d_{0}}\in \bigcap\limits_{i = 1}^{n}T_{d_{i}}\subseteq \bigcap\limits_{i = 1}^{n}\overline{T_{d}}_{i}^{\omega _{s}} $. This ends the proof that $ \mathcal{A} $ has the finite intersection property.

    Since $ \left(Y, \sigma \right) $ is $ \omega _{s} $-compact, then by Claim 1 and Theorem 5.8, there exists $ y\in \bigcap\limits_{d\in D}\overline{T_{d}} ^{\omega _{s}} $.

    Claim 2. $ y $ is an $ \omega _{s} $-cluster point of $ \left(y_{d}\right) _{d\in D} $ in $ \left(Y, \sigma \right) $.

    Proof of Claim 2. Let $ V\in \omega _{s}\left(Y, \sigma \right) $ such that $ y\in V $, and let $ d\in D $. Since $ y\in V\cap \overline{T_{d}} ^{\omega _{s}} $, then $ V\cap T_{d}\neq \emptyset $, and so there exists $ d^{\prime }\in D $ such that $ d\leq d^{\prime } $ and $ x_{d^{\prime }}\in V $. This shows that $ y $ is an $ \omega _{s} $-cluster point of $ \left(y_{d}\right) _{d\in D} $ in $ \left(Y, \sigma \right) $.

    Sufficiency. Suppose that every net in $ \left(Y, \sigma \right) $ has an $ \omega _{s} $-cluster point. We will apply Theorem 5.8. Let $ \mathcal{A} $ be a family of $ \omega _{s} $-closed sets which has the finite intersection property. Let $ D $ be the family of all finite intersections of members of $ \mathcal{A} $. Define the relation $ \leq $ on $ D $ as follows:

    $ For every d1,d2Dd1d2 if and only if d2d1.
    $

    Then $ \left(D, \leq \right) $ is a directed set. For every $ d\in D $, choose $ y_{d}\in d $. By assumption, there is an $ \omega _{s} $-cluster point $ y $ of $ \left(y_{d}\right) _{d\in D} $.

    Claim 3. $ y\in \overline{A}^{\omega _{s}} $ for all $ A\in \mathcal{A} $, and hence $ y\in \bigcap\limits_{A\in \mathcal{A}}\overline{A}^{\omega _{s}} = \bigcap\limits_{A\in \mathcal{A}}A $.

    Proof of Claim 3. Let $ A\in \mathcal{A} $ and $ V\in \omega _{s}\left(Y, \sigma \right) $ with $ y\in V $. Let $ d = A $, then $ d\in D $. Since $ y $ is an $ \omega _{s} $-cluster point of $ \left(y_{d}\right) _{d\in D} $, then there is $ d^{\prime }\in D $ such that $ d\leq d^{\prime } $ and $ y_{d^{\prime }}\in V $, say $ d^{\prime } = F $. Then $ F\subseteq A $, and hence $ y_{d^{\prime }}\in V\cap A $. Therefore, $ y\in \overline{A}^{\omega _{s}} $.

    Theorem 5.11. Let $ g:\left(Y, \sigma \right) \longrightarrow (Z, \gamma) $ be $ \omega _{s} $-irresolute and surjective. If $ \left(Y, \sigma \right) $ is $ \omega _{s} $-compact, then $ (Z, \gamma) $ is $ \omega _{s} $-compact.

    Proof. Suppose that $ \left(Y, \sigma \right) $ is $ \omega _{s} $-compact and let $ \mathcal{H} $ be a cover of $ Z $ such that $ \mathcal{H} \subseteq \omega _{s}(Z, \gamma) $. Let $ \mathcal{M} = \left\{ g^{-1}\left(H\right) :H\in \mathcal{H}\right\} $. Then $ \mathcal{M} $ is a cover of $ Y $. Also, by $ \omega _{s} $-irresoluteness of $ g $, we have $ \mathcal{M}\subseteq \omega _{s}\left(Y, \sigma \right) $. Since $ \left(Y, \sigma \right) $ is $ \omega _{s} $-compact, then there exist $ H_{1}, H_{2}, ..., H_{n}\in \mathcal{H} $ such that $ \bigcup\limits_{i = 1}^{n}g^{-1}\left(H_{i}\right) = Y $, and so $ \bigcup\limits_{i = 1}^{n}H_{i}\subseteq g\left(g^{-1}\left(\bigcup\limits_{i = 1}^{n}H_{i}\right) \right) = g\left(Y\right) $. Since $ g $ is surjective, then $ g\left(Y\right) = Z $. Therefore, $ Z = \bigcup \limits_{i = 1}^{n}H_{i} $. Hence, $ (Z, \gamma) $ is $ \omega _{s} $-compact.

    In this paper, we introduce $ \omega _{s} $-irresoluteness, $ \omega _{s} $ -openness, pre-$ \omega _{s} $-openness, and slight $ \omega _{s} $-continuity as new classes of functions. And, we define $ \omega _{s} $-compactness as a new class of topological spaces which lies between the classes compactness and semi-compactness. Several implications, examples, counter-examples, characterizations, and mapping theorems are introduced. The following topics could be considered in future studies: (1) To define $ \omega _{s} $-open separation axioms; (2) To define $ \omega _{s} $-connectedness; (3) To improve some known topological results.

    We declare no conflicts of interest in this paper.

    [1] Baillargeon R, Carey S (2012) Core cognition and beyond: The acquisition of physical and numerical knowledge. In S. Pauen (Ed.) Early childhood development and later outcome. Cambridge, England: Cambridge University Press, 33-65.
    [2] Carey S (2011) The Origin of Concepts: A Precis. Behav Brain Sci 34: 113-162
    [3] Spelke ES, Kinzler KD (2007) Core knowledge. Dev Sci 10: 89-96. doi: 10.1111/j.1467-7687.2007.00569.x
    [4] van der Lely HKJ (2005) Domain-specific cognitive systems: Insight from grammatical specific language impairment. Trends Cog Sci 9: 53-59. doi: 10.1016/j.tics.2004.12.002
    [5] Van der Lely JKJ, Pinker S (2014) The biological basis of language: insights from developmental grammatical impairments. Trends Cog Sci 18: 586-595. doi: 10.1016/j.tics.2014.07.001
    [6] Elman JL, Bates E, Johnson MH, et al. (1996) Rethinking Innateness: A Connectionist Perspective on Development. Cambridge, Mass: MIT Press.
    [7] Kirkham NZ, Slemmer JA, Johnson SP (2002) Visual statistic learning in infancy: evidence for a domain-general learning mechanism. Cognition, 83: B35-B42. doi: 10.1016/S0010-0277(02)00004-5
    [8] Lany J, Saffran JR (2013) Statistical learning mechanisms in infancy. In: Comprehensive Developmental Neuroscience: Neural Circuit Development and Function in the Brain, Volume 3 (Rubenstein JLR, Rakic P, eds). Amsterdam, Elsevier: 231-248.
    [9] Pinker S (2002) The Blank Slate. The modern denial of human nature. New York, NY: Penguin Group.
    [10] Cosmides L, Barrett HC, Tooby J (2010) Adaptative specializations, social exchange, and the evolution of human intelligence. Proc Natl Acad Sci 107: 9007-9014. doi: 10.1073/pnas.0914623107
    [11] Agrillo C, Ranpura A, Butterworth B (2010) Time and numerosity estimation are independent: Behavioral evidence for two different systems using a conflict paradigm. Cog Neurosci 1: 96-101. doi: 10.1080/17588921003632537
    [12] Clahsen H, Temple C (2003) Words and rules in children with Williams syndrome. In: Language competence across populations (Levy Y, Schaeffer J eds). Erlbaum: Mahwah, NJ, 323-352.
    [13] Cohen Kadosh R, Bahrami B, Walsh V, et al. (2011) Specialization in the human brain: The case of numbers. Front Hum Neurosci 5: 62.
    [14] Bates E, Elman J, Johnson MH, Karmiloff-Smith A, et al. (1998) Innateness and Emergentism. In A Companion to Cognitive Science (Bechtel W, Graham G, eds). Oxford: Basil Blackwell, 590-601.
    [15] Karmiloff-Smith A, Plunkett K, Johnson M, et al. (1998) What does it mean to claim that something is ‘innate'? Mind Lang 13: 588-597. doi: 10.1111/1468-0017.00095
    [16] Hyde DC, Spelke ES (2011) Neural signatures of number processing in human infants: Evidence for two core systems underlying numerical cognition. Dev Sci 14: 360-371. doi: 10.1111/j.1467-7687.2010.00987.x
    [17] Lee SA, Spelke ES (2010) A modular geometric mechanism for reorientation in children. Cognitive Psychology 61: 152-176. doi: 10.1016/j.cogpsych.2010.04.002
    [18] Spelke ES, Kinzler KD (2007) Core Knowledge. Dev Sci 10: 89-96. doi: 10.1111/j.1467-7687.2007.00569.x
    [19] Xu F, Spelke ES, Goddard S (2005) Number sense in human infants. Dev Sci 8: 88-101. doi: 10.1111/j.1467-7687.2005.00395.x
    [20] Dehaene S (1997) The number sense: how the mind creates mathematics. Oxford: Oxford University Press.
    [21] Butterworth B (1999) The mathematical brain. Macmillan: London.
    [22] Gelman R, Gallistel R (1986) The child's understanding of number. Cambridge, MA: Harvard University Press.
    [23] Duchaine B (2000) Developmental propagnosia with normal configural processing. NeuroReport 11: 79-83. doi: 10.1097/00001756-200001170-00016
    [24] Duchaine B, Nakayama K (2005) Dissociations of face and object recognition in developmental prosopagnosia. J Cog Neurosci 17: 249-261. doi: 10.1162/0898929053124857
    [25] Duchaine B, Nakayama K (2006) Developmental prosopagnosia: a window to content-specific face processing. Cur Op Neurobio 16: 166-173. doi: 10.1016/j.conb.2006.03.003
    [26] Duchaine B, Yovel G, Butterworth E, et al. (2006) Prosopagnosia as an impairment to face-specific mechanisms: elimination of the alternative explanations in a developmental case. Cognitive Neuropsychology 23: 714-747. doi: 10.1080/02643290500441296
    [27] Landau B, Hoffman JE, Kurz N (2005) Object definitions with severe spatial deficits in Williams syndrome: sparing and breadkdown. Cognition 100: 483-510.
    [28] Piattelli-Palmarini M (2001) Speaking of learning: How do we acquire our marvellous facility for expressing ourselves in words? Nature 411: 887-888. doi: 10.1038/35082123
    [29] Spelke ES, Kinzler KD (2009) Innateness, learning and rationality. Child Dev Pers 3: 96-98. doi: 10.1111/j.1750-8606.2009.00085.x
    [30] Duchaine B, Cosmides L, Tooby J (2001) Evolutionary psychology and the brain. Curr Op Neurobio 11: 225-230. doi: 10.1016/S0959-4388(00)00201-4
    [31] Hauser MD, Spelke ES (2004). Evolutionary and developmental foundations of human knowledge: A case study of mathematics. In M Gazzaniga (Ed.) The Cognitive Neurosciences, Vol. 3. Cambridge: MIT Press, 853-864.
    [32] Bolhuis JJ, Brown GR, Richardson RC, et al. (2011) Darwin in Mind: New Opportunities for Evolutionary Psychology. PLoS Biology 9: 1-8.
    [33] Baron-Cohen S, Leslie AM, Frith U (1985) Does the autistic child have a “theory of mind”? Cognition 21: 37-46. doi: 10.1016/0010-0277(85)90022-8
    [34] Leslie AM (1992) Pretence, autism, and the theory-of-mind-module. Cur Dir Psych Sci 1: 18-21. doi: 10.1111/1467-8721.ep10767818
    [35] Baron-Cohen S, Ring HA, Wheelwright S, et al. (1999) Social intelligence in the normal and autistic brain: an fMRI study. Eur J Neurosci 11: 1891-8. doi: 10.1046/j.1460-9568.1999.00621.x
    [36] Temple CM (1997) Cognitive neuropsychology and its application to children. J Child Psychol Psychiatry 38: 27-52. doi: 10.1111/j.1469-7610.1997.tb01504.x
    [37] Butterworth B (2008) State-of-science review: Dyscalculia. In Goswami UC (Ed.) Foresight Mental Capital and Mental Wellbeing. Office of Science and Innovation, London.
    [38] Duchaine B, Nieminen-von Wendt T, New J, et al. (2003) Dissociations of visual recognition in a developmental agnosic: evidence for separate developmental processes. Neurocase 9: 380-389. doi: 10.1076/neur.9.5.380.16556
    [39] Molko N, Cachia A, Rivière D, et al. (2003) Functional and Structural Alterations of the Intraparietal Sulcus in a Developmental Dyscalculia of Genetic Origin. Neuron 40: 847-858. doi: 10.1016/S0896-6273(03)00670-6
    [40] Rice M (1999) Specific grammatical limitations in children with Specific Language Impairment. In: Neurodevelopmental disorders (Tager-Flusberg H, ed). Cambridge, Mass: MIT Press, 331-360.
    [41] Shalev RS, Manor O, Gross-Tsur V (2005) Developmental dyscalculia: a prospective six-year follow-up. Dev Med Child Neurol 2: 121-125.
    [42] Castle A, Coltheart M (1993) Varieties of developmental dyslexia. Cognition 47: 149-180. doi: 10.1016/0010-0277(93)90003-E
    [43] Gopnik M (1990) Genetic basis of grammar defect. Nature 347: 26-26.
    [44] Karmiloff-Smith A (1992) Beyond Modularity: A Developmental Perspective on Cognitive Science. Cambridge, Mass: MIT Press/Bradford Books.
    [45] Karmiloff-Smith A (1998) Development itself is the key to understanding developmental disorders. Trends Cog Sci 2: 389-398. doi: 10.1016/S1364-6613(98)01230-3
    [46] Mareschal D, Johnson MH, Sirios S, et al. (2007) Neuroconstructivism: Vol. I. How the brain constructs cognition. Oxford, England: Oxford University Press.
    [47] Westerman G, Marescal D, Johnson MH, et al. (2007) Neuroconstructivism. Dev Sci 10: 75-83. doi: 10.1111/j.1467-7687.2007.00567.x
    [48] Westermann G, Thomas MSC, Karmiloff-Smith A (2010) Neuroconstructivism. In: Handbook of Childhood Development (Goswami U, ed). Oxford: Wiley-Blackwell, 723-748.
    [49] Karmiloff-Smith A (2013) Challenging the use of adult neuropsychological models for explaining neurodevelopmental disorders: Developed versus developing brains. Quart J Exp Psych 66: 1-14. doi: 10.1080/17470218.2012.744424
    [50] Paterson SJ, Brown JH, Gsödl MK, et al. (1999) Cognitive Modularity and Genetic Disorders, Science 286: 2355-2358.
    [51] Johnson MH (2001) Functional brain development in humans. Nat Rev Neurosci 2: 475-483. doi: 10.1038/35081509
    [52] Stiles J (2009) The Fundamentals of Brain Development. Cambridge MA: Havard University Press.
    [53] Thomas MSC, Knowland VC, Karmiloff-Smith A (2011) Mechanisms of developmental regression in autism and the broader phenotype: a neural network modeling approach. Psych Rev 118: 637-654. doi: 10.1037/a0025234
    [54] Casey BJ, Giedd JN, Thomas KM (2000) Structural and functional brain development and its relation to cognitive development. Biol Psychol 54: 241-257. doi: 10.1016/S0301-0511(00)00058-2
    [55] Huttenlocher PR, de Courten C (1987) The development of synapses in striate cortex of man. Hum Neurobio 6: 1-9.
    [56] Huttenlocher PR, Dabholkar AS (1997) Regional Differences in Synaptogenesis in Human. J Comp Neurol 387: 167-178. doi: 10.1002/(SICI)1096-9861(19971020)387:2<167::AID-CNE1>3.0.CO;2-Z
    [57] Giedd J, Blumenthal J, Jeffries N, et al. (1999) Brain development during childhood and adolescence: A longitudinal MRI study. Nat Neurosci 2: 861-863. doi: 10.1038/13158
    [58] Giedd J, Rumsey J, Castellanos F, et al. (1996) A quantitative MRI study of the corpus callosum in children and adolescents. Dev Brain Res 91: 274-280. doi: 10.1016/0165-3806(95)00193-X
    [59] Cohen Kadosh K., Henson RN, Cohen Kadosh R, et al. (2009) Task-dependent activation of face-sensitive cortex: an fMRI adaptation study. J Cog Neurosci, 22: 903-917.
    [60] De Haan M, Humphreys K, Johnson MH (2002) Developing a brain specializes for face processing: A converging methods approach. Dev Psychobiology 40: 200-212. doi: 10.1002/dev.10027
    [61] Krishnan S, Leech R, Mercure E, et al. (2014) Convergent and divergent fMRI responses in children and adults to increasing language production demands. Cereb Cortex pii: bhu120.
    [62] Mills DL, Coffy-Corins S, Neville H (1997) Language comprehension and cerebral specialisation from 13-20 months. Dev Psych 13: 397-445.
    [63] Minagawa-Kawai Y, Mori K, Naoi N, et al. (2007) Neural attunement processes in infants during the acquisition of language-specific phonemic contrasts. J Neurosci 3: 315-321.
    [64] Neville H, Mills D, Bellugi U (1994) Effects of altered auditory sensitivity and age of language acquisition on the development of language-relevant neural systems: Preliminary studies of William syndrome. In: Atypical cognitive deficits in developmental disorders: Implications for brain function (Broman S, Grafman J, eds). Hillsdale, NJ: Erlbaum, 67-83.
    [65] Stiles J (2012) Neural plasticity and cognitive development: Insights from children with perinatal brain injury. Oxford UK: Oxford University Press.
    [66] Bates E, Roe K (2001) Language development in children with unilateral brain injury. In: Handbook of Developmental Cognitive Neuroscience (Nelson CA, Luciana M, eds). Cambridge, MA: MIT Press, 281-307.
    [67] Casey BJ, Tottenham N, Liston C, et al. (2005) Imaging the developing brain. What have we learned about cognitive development? Trends Cog Sci 9: 104-110.
    [68] Durston S, Davidson MC, Tottenham N, et al. (2006). A shift from diffuse to focal cortical activity with development. Dev Sci 9: 1-8.
    [69] Paterson SJ, Heim S, Friedman JT, et al. (2006) Development of structure and function in the infant brain: Implications for cognition, language and social behaviour. Neurosci Biobehav Rev 30: 1087-1105. doi: 10.1016/j.neubiorev.2006.05.001
    [70] Smith LB, Thelen E (2003) Development as a dynamic system. Trends Cog Sci 7: 343-348. doi: 10.1016/S1364-6613(03)00156-6
    [71] Tyler LK, Shafto MA, Randall B, et al. (2010) Preserving Syntactic Processing across the Adult Life Span: The Modulation of the Frontotemporal Language System in the Context of Age-Related Atrophy. Cereb Cortex 20: 352-364. doi: 10.1093/cercor/bhp105
    [72] Dehaene S, Cohen L (2007) Cultural recycling of cortical maps. Neuron 56: 384-398. doi: 10.1016/j.neuron.2007.10.004
    [73] Fodor J (1983) Modularity of Mind. Cambridge, MA: MIT Press.
    [74] Dehaene S, Charles L, King JR, et al. (2014) Toward a computational theory of conscious processing. Curr Op Neurobio 25: 76-84. doi: 10.1016/j.conb.2013.12.005
    [75] Anderson ML (2010) Neural reuse: A fundamental organizational principle of the brain. Behav Brain Sci 33: 245-313. doi: 10.1017/S0140525X10000853
    [76] Dehaene S (2009) Reading in the brain: The science and evolution of a human invention. New York, NY: Penguin Group
    [77] Goldman-Rakic PS (1987) Circuitry of the pre- frontal cortex and the regulation of behavior by representational knowledge. In: Handbook of Physiology 5 (Plum F, Mountcastle V, eds). Bethesda, MD: American Physiological Society, 373-417.
    [78] Johnson MH (2004) Plasticity and functional brain development: The case of face processing. In: Attention & Performance XX: Functional neuroimaging of visual cognition (Kanwisher N, Duncan J eds). Oxford: Oxford University Press, 257- 263.
    [79] Kuhl P (2004) Early language acquisition: cracking the speech code. Nat Rev Neurosci 5: 831-843. doi: 10.1038/nrn1533
    [80] Majdan M, Schatz CJ (2006) Effects of visual experience on activity-dependent gene regulation in cortex. Nat Neurosci 9: 650-659. doi: 10.1038/nn1674
    [81] Meaney MJ, Szyf M (2005) Maternal care as a model for experience-dependent chromatin plasticity? Trends Neurosci 28: 456-463. doi: 10.1016/j.tins.2005.07.006
    [82] Bhattacharya J, Petsche H (2005) Drawing on mind's canvas: Differences in cortical integration patterns between artists and non-artists. Hum Brain Mapping 26: 1-14. doi: 10.1002/hbm.20104
    [83] Crone EA, Elzinga B (2014) Changing brains: How longitudinal fMRI studies can inform us about cognitive and social-affective growth trajectories. WIREs Cogn Sci 6: 53-63.
    [84] Brown TT, Petersen SE, Schlaggar BL (2006) Does human functional brain organization shift from different to focal with development? Dev Sci 9: 9-10. doi: 10.1111/j.1467-7687.2005.00455.x
    [85] Karmiloff-Smith A (1997) Promissory notes, genetic clocks or epigenetic outcomes? Behav Brain Sci 20: 359-377.
    [86] Karmiloff-Smith A (2007) Atypical epigenesis. Dev Sci 10: 84-88. doi: 10.1111/j.1467-7687.2007.00568.x
    [87] Kingsbury MA, Finlay LF (2001) The cortex in multidimensional space: where do cortical areas come from? Developmental Science 4: 125-142. doi: 10.1111/1467-7687.00158
    [88] Bishop KM, Goudreau G, O'Leary DDM (2000) Regulation of area identity in the mammalian neocortex by Emx2 and Pax6. Science 288: 344-349. doi: 10.1126/science.288.5464.344
    [89] Kaffman A, Meaney MJ (2007) Neurodevelopmental sequelae of postnatal maternal care in rodents: clinical and research implications of molecular insights. J Child Psychol. Psychiatry 48: 224-244. doi: 10.1111/j.1469-7610.2007.01730.x
    [90] Gotlieb, G (2007) Probabilistic epigenesis. Dev Sci 10: 1-11. doi: 10.1111/j.1467-7687.2007.00556.x
    [91] Scerif G, Karmiloff-Smith A (2001) Genes and environment: What does interaction really mean? Trends Gen 17: 418-419. doi: 10.1016/S0168-9525(01)02337-X
    [92] Quartz S, Sejnowski T (1997) The neural basis of cognitive development: A constructivist manifesto. Behav Brain Sci 20: 537-596.
    [93] Finlay BL (2007) E pluribus unum: Too many unique human capacities and too many theories. In: The evolution of mind: Fundamental questions and controversies (Gangestad S, Simpson J, eds). New York: Guilford Press, 294-304.
    [94] Karmiloff-Smith A (2010) Neuroimaging of the developing brain: Taking “developing” seriously. Hum Brain Map 31: 934-941. doi: 10.1002/hbm.21074
    [95] Karmiloff-Smith A (2012) Brain: The Neuroconstructivist Approach. In: Neuro- developmental disorders across the lifespan: A neuroconstructivist approach (Farran EK, Karmiloff-Smith A, eds) Oxford: Oxford University Press, 37-58.
    [96] Karmiloff-Smith A, Thomas MSC (2004) Can developmental disorders be used to bolster claims from Evolutionary Psychology? A neuroconstructivist approach. In: Biology and Knowledge Revisited: From Neurogenesis to Psychogenesis (Langer J, Taylor Parker S, Milbrath C, eds) Mahwah, NJ: Lawrence Erlbaum Associates, 307-322.
    [97] Karmiloff-Smith A (2009) Nativism versus Neuroconstructivism: Rethinking the Study of Developmental Disorders. Interplay of Biology and Environment, Dev Psych 45: 56-63.
    [98] Pinker S (1994) The language instinct: How the mind creates language. New York: W.Morrow.
    [99] Pinker S (1999) Words and rules: The ingredients of language. New York: Basic Books.
    [100] Karmiloff-Smith A, Thomas MSC, Annaz D, et al. (2004) Exploring the Williams Syndrome Face Processing Debate: The importance of building developmental trajectories. J Child Psych Psychiatry 45: 1258-1274. doi: 10.1111/j.1469-7610.2004.00322.x
    [101] Karmiloff-Smith A, D'Souza D, Dekker TM, et al. (2012) Genetic and environmental vulnerabilities in children with neurodevelopmental disorders. Proc Natl Acad Sci 109: 17261-5. doi: 10.1073/pnas.1121087109
    [102] D'Souza D, Karmiloff-Smith A (2011) When modularization fails to occur: a developmental perspective. Cog Neuropsych 28: 276-287. doi: 10.1080/02643294.2011.614939
    [103] Mackinlay R, Charman T, Karmiloff-Smith A (2006) High functioning children with autism spectrum disorder: A novel test of multitasking. Brain Cog 61: 14-24. doi: 10.1016/j.bandc.2005.12.006
    [104] Comery TA, Harris JB, Willems PJ, et al. (1997) Abnormal dendritic spines in fragile X knockout mice: Maturation and pruning deficits. Proc Natl Acad Sci 94: 5401-5404. doi: 10.1073/pnas.94.10.5401
    [105] Oliver A, Johnson MH, Karmiloff-Smith A, et al. (2000) Deviations in the emergence of representations: A neuroconstructivist framework for analysing developmental disorders. Dev Sci 3: 1-23. doi: 10.1111/1467-7687.00094
    [106] Karmiloff-Smith A, Scerif G, Ansari D (2003) Double dissociations in developmental disorders? Theoretically misconceived, empirically dubious. Cortex 39: 161-163.
    [107] Johnson MH, Halit H, Grice S, et al. (2002) Neuroimaging of typical and atypical development: A perspective from multiple levels of analysis. Dev Psychopath 14: 521-536.
    [108] Cicchetti D, & Tucker D (1994) Development and self-regulatory structures of the mind. Dev Psychopath 6: 533-549. doi: 10.1017/S0954579400004673
    [109] Sur M, Pallas SL, Roe AW (1990) Crossmodal plasticity in cortical development: Differentiation and specification of sensory neocortex. TINS 13: 227-233.
    [110] Webster MJ, Bachevalier J, Ungerleider LG (1995) Development and plasticity of visual memory circuits. In: Maturational windows and adult cortical plasticity in human development: Is there reason for an optimistic view? (Julesz B, Kovacs I, eds). Reading, MA: Addison-Wesley.
    [111] Bavelier D, Levi DM, Li RW, et al. (2010) Removing brakes on adult plasticity: from molecular to behavioural interventions. J Neurosci 30: 14964-71. doi: 10.1523/JNEUROSCI.4812-10.2010
    [112] Hensch TK (2005) Critical period plasticity in local cortical circuits. Nat. Rev. Neurosci 6: 877-888.
    [113] Karmiloff-Smith A (1994) Transforming a partially structured brain into a creative mind. Behav Brain Sci 17: 732-745. doi: 10.1017/S0140525X00036906
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