Soft set theory is a theme of interest for many authors working in various areas because of its rich potential for applications in many directions. it received the attention of the topologists who always seeking to generalize and apply the topological notions on different structures. To contribute to this research area, in this paper, we formulate new soft separation axioms, namely tt-soft αTi(i=0,1,2,3,4) and tt-soft α-regular spaces. They are defined using total belong and total non-belong relations with respect to ordinary points. This way of definition helps us to generalize existing comparable properties via general topology and to remove a strict condition of the shape of soft open and closed subsets of soft α-regular spaces. With the help of examples, we show the relationships between them as well as with soft αTi(i=0,1,2,3,4) and soft α-regular spaces. Also, we explore under what conditions they are kept between soft topological space and its parametric topological spaces. We characterize tt-soft αT1 and tt-soft α-regular spaces and give some conditions that guarantee the equivalence of tt-soft αTi(i=0,1,2) and the equivalence of tt-soft αTi(i=1,2,3). Further, we investigate some interrelations of them and some soft topological notions such as soft compactness, product soft spaces and sum of soft topological spaces. In the end, we study the main properties of of α-fixed soft point theorem.
Citation: Tareq M. Al-shami, El-Sayed A. Abo-Tabl. Soft α-separation axioms and α-fixed soft points[J]. AIMS Mathematics, 2021, 6(6): 5675-5694. doi: 10.3934/math.2021335
[1] | Tareq M. Al-shami, Zanyar A. Ameen, A. A. Azzam, Mohammed E. El-Shafei . Soft separation axioms via soft topological operators. AIMS Mathematics, 2022, 7(8): 15107-15119. doi: 10.3934/math.2022828 |
[2] | Tareq M. Al-shami, Abdelwaheb Mhemdi . A weak form of soft $ \alpha $-open sets and its applications via soft topologies. AIMS Mathematics, 2023, 8(5): 11373-11396. doi: 10.3934/math.2023576 |
[3] | Samer Al-Ghour, Hanan Al-Saadi . Soft weakly connected sets and soft weakly connected components. AIMS Mathematics, 2024, 9(1): 1562-1575. doi: 10.3934/math.2024077 |
[4] | Orhan Göçür . Amply soft set and its topologies: AS and PAS topologies. AIMS Mathematics, 2021, 6(4): 3121-3141. doi: 10.3934/math.2021189 |
[5] | Tareq M. Al-shami, Salem Saleh, Alaa M. Abd El-latif, Abdelwaheb Mhemdi . Novel categories of spaces in the frame of fuzzy soft topologies. AIMS Mathematics, 2024, 9(3): 6305-6320. doi: 10.3934/math.2024307 |
[6] | Dina Abuzaid, Samer Al-Ghour . Supra soft Omega-open sets and supra soft Omega-regularity. AIMS Mathematics, 2025, 10(3): 6636-6651. doi: 10.3934/math.2025303 |
[7] | Tareq M. Al-shami, Abdelwaheb Mhemdi, Radwan Abu-Gdairi, Mohammed E. El-Shafei . Compactness and connectedness via the class of soft somewhat open sets. AIMS Mathematics, 2023, 8(1): 815-840. doi: 10.3934/math.2023040 |
[8] | Arife Atay . Disjoint union of fuzzy soft topological spaces. AIMS Mathematics, 2023, 8(5): 10547-10557. doi: 10.3934/math.2023535 |
[9] | Mesfer H. Alqahtani, Zanyar A. Ameen . Soft nodec spaces. AIMS Mathematics, 2024, 9(2): 3289-3302. doi: 10.3934/math.2024160 |
[10] | Rehab Alharbi, S. E. Abbas, E. El-Sanowsy, H. M. Khiamy, Ismail Ibedou . Soft closure spaces via soft ideals. AIMS Mathematics, 2024, 9(3): 6379-6410. doi: 10.3934/math.2024311 |
Soft set theory is a theme of interest for many authors working in various areas because of its rich potential for applications in many directions. it received the attention of the topologists who always seeking to generalize and apply the topological notions on different structures. To contribute to this research area, in this paper, we formulate new soft separation axioms, namely tt-soft αTi(i=0,1,2,3,4) and tt-soft α-regular spaces. They are defined using total belong and total non-belong relations with respect to ordinary points. This way of definition helps us to generalize existing comparable properties via general topology and to remove a strict condition of the shape of soft open and closed subsets of soft α-regular spaces. With the help of examples, we show the relationships between them as well as with soft αTi(i=0,1,2,3,4) and soft α-regular spaces. Also, we explore under what conditions they are kept between soft topological space and its parametric topological spaces. We characterize tt-soft αT1 and tt-soft α-regular spaces and give some conditions that guarantee the equivalence of tt-soft αTi(i=0,1,2) and the equivalence of tt-soft αTi(i=1,2,3). Further, we investigate some interrelations of them and some soft topological notions such as soft compactness, product soft spaces and sum of soft topological spaces. In the end, we study the main properties of of α-fixed soft point theorem.
In 1999, Molodtsov [30] founded a novel mathematical tool for dealing with uncertainties, namely soft set. One of the merits of this tool is its free from the difficulties that the other existing methods such as fuzzy set theory and probability theory. This matter makes soft set theory very popular research area all over the globe. Immediately afterwards, Maji et al. [26] in 2003, established the basis of soft operations between soft sets. Although some of these operations were considered ill-defined, they formed the starting point of constructing soft set theory. In this regard, Ali et al. [5] redefined some soft operators to make them more functional for improving several new results and they explored new soft operators such as restricted union and restricted intersection of two soft sets.
In 2011, Shabir and Naz [32] exploited soft sets to introduce soft topological spaces. The fundamental soft topological notions such as the operators of soft closure and interior, soft subspace and soft separation axioms were investigated by them. Min [29] completed study of soft separation axioms and revised some results obtained in [32]. Soft compactness was introduced and discussed by Aygünoǧlu and Aygün [14] in 2012. Hida [23] defined another type of soft compactness depending on the belong relations. Al-shami [10] did some amendments concern some types of soft union and intersection. Then, he [11] studied new types of soft compactness. The authors of [25] presented soft maps by using two crisp maps, one of them between the sets of parameters and the second one between the universal sets. However, the authors of [36] introduced soft maps by using the concept of soft points. Some applications of different types of soft maps were the goal of some articles, see [25,27,36].
Until 2018, the belong and non-belong relations that utilized in these studies are those given by [32]. In 2018, the authors of [20] came up new relations of belong and non-belong between an element and soft set, namely partial belong and total non-belong relations. In fact, these relations widely open the door to study and redefine many soft topological notions. This leads to obtain many fruitful properties and changes which can be seen significantly on the study of soft separation axioms as it was showed in [9,20,21]. As another path of study soft separation axioms, the authors of [16,35] studied them with respect to the distinct soft points. Recently, some applications of compactness and soft separation axioms have been investigated in [6,7,33,34].
Das and Samanta [19] studied the concept of a soft metric based on the soft real set and soft real numbers given in [18]. Wardowski [36] tackled the fixed point in the setup of soft topological spaces. Abbas et al.[1] presented soft contraction mappings and established a soft Banach fixed point theorem in the framework of soft metric spaces. Recently, many researchers explored fixed point findings in soft metric type spaces, see, for example, [2,37]. Some interesting works regarding deferential equations were given in [17,24,28].
One of the significant ideas that helps to prove some properties and remove some problems on soft topology is the concept of a soft point. It was first defined by Zorlutuna et al. [38] in order to study interior points of a soft set and soft neighborhood systems. Then [19] and [31] simultaneously redefined soft points to discuss soft metric spaces. In fact, the recent definition of a soft point makes similarity between many set-theoretic properties and their counterparts on soft setting. Two types of soft topologies, namely enriched soft topology and extended soft topology were studied in [14] and [31], respectively. The equivalence between these two topologies have been recently proved by Al-shami and Kočinac [12].
We organized this paper as follows: After this introduction, we allocate Section (2) to recall some definitions and results of soft sets and soft topologies that will help us to understand this work. Section (3) introduces tt-soft αTi(i=0,1,2,3,4) and tt-soft α-regular spaces with respect to ordinary points by using total belong and total non-belong relations. The relationships between them and their main properties are discussed with the help of interesting examples. In Section (4), we explore an α-fixed soft point theorem and study some main properties. In particular, we conclude under what conditions α-fixed soft points are preserved between a soft topological space and its parametric topological spaces. Section (6) concludes the paper.
To well understand the results obtained in this study, we shall recall some basic concepts, definitions and properties from the literature.
Definition 2.1. [30] For a nonempty set X and a set of parameters E, a pair (G,E) is said to be a soft set over X provided that G is a map of E into the power set P(X).
In this study, we use a symbol GE to refer a soft set instead of (G,E) and we identify it as ordered pairs GE={(e,G(e)):e∈E and G(e)∈P(X)}.
A family of all soft sets defined over X with E is denoted by S(XE).
Definition 2.2. [22] A soft set GE is said to be a subset of a soft set HE, denoted by GE˜⊆HE, if G(e)⊆H(e) for each e∈E.
The soft sets GE and HE are said to be soft equal if each one of them is a subset of the other.
Definition 2.3. [20,32] Let GE be a soft set over X and x∈X. We say that:
(ⅰ) x∈GE, it is read: x totally belongs to GE, if x∈G(e) for each e∈E.
(ⅱ) x∉GE, it is read: x does not partially belong to GE, if x∉G(e) for some e∈E.
(ⅲ) x⋐GE, it is read: x partially belongs to GE, if x∈G(e) for some e∈E.
(ⅳ) x⋐̸GE, it is read: x does not totally belong to GE, if x∉G(e) for each e∈E.
Definition 2.4. Let GE be a soft set over X and x∈X. We say that:
(ⅰ) GE totally contains x if x∈GE.
(ⅱ) GE does not partially contain x if x∉GE.
(ⅲ) GE partially contains x if x⋐GE.
(ⅳ) GE does not totally contain x if x⋐̸GE.
Definition 2.5. [5] The relative complement of a soft set GE is a soft set GcE, where Gc:E→2X is a mapping defined by Gc(e)=X∖G(e) for all e∈E.
Definition 2.6. [19,20,26,31] A soft set (G,E) over X is said to be:
(ⅰ) a null soft set, denoted by ˜Φ, if G(e)=∅ for each e∈E.
(ⅱ) an absolute soft set, denoted by ˜X, if G(e)=X for each e∈E.
(ⅲ) a soft point Pxe if there are e∈E and x∈X such that G(e)={x} and G(e′)=∅ for each e′∈E∖{e}. We write that Pxe∈GE if x∈G(e).
(ⅳ) a stable soft set, denoted by ˜S, if there is a subset S of X such that G(e)=S for each e∈E. In particular, we denoted by xE if S={x}.
(ⅴ) a countable (resp. finite) soft set if G(e) is countable (resp. finite) for each e∈E. Otherwise, it is said to be uncountable (resp. infinite).
Definition 2.7. [5,26] Let GE and HE be two soft sets over X.
(ⅰ) Their intersection, denoted by GE˜⋂HE, is a soft set UE, where a mapping U:E→2X is given by U(e)=G(e)⋂H(e).
(ⅱ) Their union, denoted by GE˜⋃HE, is a soft set UE, where a mapping U:E→2X is given by U(e)=G(e)⋃H(e).
Definition 2.8. [15] Let GE and HF be two soft sets over X and Y, respectively. Then the cartesian product of GE and HF, denoted by G×HE×F, is defined as (G×H)(e,f)=G(e)×H(f) for each (e,f)∈E×F.
The soft union and intersection operators were generalized for any number of soft sets in a similar way.
Definition 2.9. [25] A soft mapping between S(XA) and S(YB) is a pair (f,ϕ), denoted also by fϕ, of mappings such that f:X→Y, ϕ:A→B. Let GA and HB be subsets of S(XA) and S(YB), respectively. Then the image of GA and pre-image of HB are defined as follows.
(ⅰ) fϕ(GA)=(fϕ(G))B is a subset of S(YB) such fϕ(G)(b)=⋃a∈ϕ−1(b)f(G(a)) for each b∈B.
(ⅱ) f−1ϕ(HB)=(f−1ϕ(H))A is a subset of S(XA) such that f−1ϕ(H)(a)=f−1(H(ϕ(a))) for each a∈A.
Definition 2.10. [38] A soft map fϕ:S(XA)→S(YB) is said to be injective (resp. surjective, bijective) if ϕ and f are injective (resp. surjective, bijective).
Definition 2.11. [32] A family τ of soft sets over X under a fixed set of parameters E is said to be a soft topology on X if it satisfies the following.
(ⅰ) ˜X and ˜Φ are members of τ.
(ⅱ) The intersection of a finite number of soft sets in τ is a member of τ.
(ⅲ) The union of an arbitrary number of soft sets in τ is a member of τ.
The triple (X,τ,E) is called a soft topological space. A member in τ is called soft open and its relative complement is called soft closed.
Proposition 2.12. [32] In (X,τ,E), a family τe={G(e):GE∈τ} is a classical topology on X for each e∈E.
τe is called a parametric topology and (X,τe) is called a parametric topological space.
Definition 2.13. [32] Let (X,τ,E) be a soft topological space and ∅≠Y⊆X. A family τY={˜Y˜⋂GE:GE∈τ} is called a soft relative topology on Y and the triple (Y,τY,E) is called a soft subspace of (X,τ,E).
Definition 2.14. [3] A subset GE of (X,τ,E) is called soft α-open if GE˜⊆int(cl(int(GE))).
The following result will help us to establish some properties of soft α-separation axioms and soft α-compact spaces, see, for example, Theorem (3.16) and Proposition (3.24). It implies that the family of soft α-open subsets of (X,τ,E) forms a new soft topology τα over X that is finer than τ. In fact, this characteristic of soft α-open sets that does not exist for the families of soft semi-open, soft pre-open, soft b-open and soft β-open sets.
(ⅰ) Every soft open set is soft α-open.
(ⅱ) The arbitrary union (finite intersection) of soft α-open sets is soft α-open.
Definition 2.16. [3] Let GE be a subset of (X,τ,E). Then ¯GEα is the intersection of all soft α-closed sets containing GE.
It is clear that: x∈¯GEα if and only if GE˜⋂UE≠˜Φ for each soft α-open set UE totally containing x; and Pxe∈¯GEα if and only if GE˜⋂UE≠˜Φ for each soft α-open set UE totally containing Pxe.
Proposition 2.17. [8] Let ˜Y be soft open subset of (X,τ,E). Then:
1. If (H,E) is soft α-open and ˜Y is soft open in (X,τ,E), then (H,E)˜⋂(Y,E) is a soft α-open subset of (Y,τY,E).
2. If ˜Y is soft open in (X,τ,E) and (H,E) is a soft α-open in (Y,τY,E), then (H,E) is a soft α-open subset of (X,τ,E).
Definition 2.18. [4] (X,τ,E) is said to be:
(ⅰ) soft αT0 if for every x≠y∈X, there is a soft α-open set UE such that x∈UE and y∉UE; or y∈UE and x∉UE.
(ⅱ) soft αT1 if for every x≠y∈X, there are two soft α-open sets UE and VE such that x∈UE and y∉UE; and y∈VE and x∉VE.
(ⅲ) soft αT2 if for every x≠y∈X, there are two disjoint soft α-open sets UE and VE such that x∈GE and y∈FE.
(ⅳ) soft α-regular if for every soft α-closed set HE and x∈X such that x∉HE, there are two disjoint soft α-open sets UE and VE such that HE˜⊆UE and x∈VE.
(ⅴ) soft α-normal if for every two disjoint soft α-closed sets HE and FE, there are two disjoint soft α-open sets UE and VE such that HE˜⊆UE and FE˜⊆VE.
(ⅵ) soft αT3 (resp. soft αT4) if it is both soft α-regular (resp. soft α-normal) and soft αT1-space.
Definition 2.19. [8] A family {GiE:i∈I} of soft α-open subsets of (X,τ,E) is said to be a soft α-open cover of ˜X if ˜X=˜⋃i∈IGiE.
Definition 2.20. [8] (X,τ,E) is said to be:
(ⅰ) soft αT′2 if for every Pxe≠Pye′∈˜X, there are two disjoint soft α-open sets UE and VE containing Pxe and Pye′, respectively.
(ⅱ) soft α-compact if every soft α-open cover of ˜X has a finite subcover.
Proposition 2.21. [8]
(ⅰ) A soft α-compact subset of a soft αT′2-space is soft α-closed.
(ⅱ) A stable soft α-compact subset of a soft αT2-space is soft α-closed.
To study the properties that preserved under soft α⋆-homeomorphism maps, the concept of a soft α-irresolute map will be presented in this work under the name of a soft α⋆-continuous map.
Definition 2.22. [8] gφ:(X,τ,E)→(Y,θ,E) is called soft α⋆-continuous if the inverse image of each soft α-open set is soft α-open.
Proposition 2.23. [8] The soft α⋆-continuous image of a soft α-compact set is soft α-compact.
Definition 2.24. [3] A soft map fφ:(X,τ,A)→(Y,θ,B) is said to be:
(ⅰ) soft α-continuous if the inverse image of each soft open set is soft α-open.
(ⅱ) soft α-open (resp. soft α-closed) if the image of each soft open (resp. soft closed) set is soft α-open (resp. soft α-closed).
(ⅲ) a soft α-homeomorphism if it is bijective, soft α-continuous and soft α-open.
Definition 2.25. A soft topology τ on X is said to be:
(ⅰ) an enriched soft topology [14] if all soft sets GE such that G(e)=∅ or X are members of τ.
(ⅱ) an extended soft topology [31] if τ={GE:G(e)∈τe for each e∈E}, where τe is a parametric topology on X.
Al-shami and Kočinac [12] proved the equivalence of enriched and extended soft topologies and obtained many useful results that help to study the relationships between soft topological spaces and their parametric topological spaces.
Theorem 2.26. [12] A subset (F,E) of an extended soft topological space (X,τ,E) is soft α-open if and only if each e-approximate element of (F,E) is α-open.
Proposition 2.27. [13] Let {(Xi,τi,E):i∈I} be a family of pairwise disjoint soft topological spaces and X=⋃i∈IXi. Then the collection
τ={(G,E)˜⊆˜X:(G,E)˜⋂~Xiisasoftopensetin(Xi,τi,E)foreveryi∈I} |
defines a soft topology on X with a fixed set of parameters E.
Definition 2.28. [13] The soft topological space (X,τ,E) given in the above proposition is said to be the sum of soft topological spaces and is denoted by (⊕i∈IXi,τ,E).
Theorem 2.29. [13] A soft set (G,E)˜⊆~⊕i∈IXi is soft α-open (resp. soft α-closed) in (⊕i∈IXi,τ,E) if and only if all (G,E)˜⋂~Xi are soft α-open (resp. soft α-closed) in (Xi,τi,E).
Proposition 2.30. [36] Let gφ:(X,τ,E)→(X,τ,E) be a soft map such that ˜⋂n∈Ngnφ(˜X) is a soft point Pxe. Then Pxe is a unique fixed point of gφ.
Theorem 2.31. [38] Let (X,τ,A) and (Y,θ,B) be two soft topological spaces and Ω={GA×FB:GA∈τ and FB∈θ}. Then the family of all arbitrary union of elements of Ω is a soft topology over X×Y under a fixed set of parameters A×B.
Lemma 2.32. [7] Let (G,A) and (H,B) be two subsets of (X1,τ1,A) and (X2,τ2,B), respectively. Then:
(ⅰ) cl(G,A)×cl(H,B)=cl((G,A)×(H,B)).
(ⅱ) int(G,A)×int(H,B)=int((G,A)×(H,B)).
This section introduces the concepts of tt-soft αTi(i=0,1,2,3,4) and tt-soft α-regular spaces, where tt denote the total belong and total non-belong relations that are utilized in the definitions of these concepts. The relationships between them are showed and their main features are studied. In addition, their behaviours with the concepts of hereditary, topological and additive properties are investigated. Some examples are provided to elucidate the obtained results.
Definition 3.1. (X,τ,E) is said to be:
(ⅰ) tt-soft αT0 if for every x≠y∈X, there exists a soft α-open set UE such that x∈UE and y⋐̸UE or y∈UE and x⋐̸UE.
(ⅱ) tt-soft αT1 if for every x≠y∈X, there exist soft α-open sets UE and VE such that x∈UE and y⋐̸UE; and y∈VE and x⋐̸VE.
(ⅲ) tt-soft αT2 if for every x≠y∈X, there exist two disjoint soft α-open sets UE and VE such that x∈UE and y⋐̸UE; and y∈VE and x⋐̸VE.
(ⅳ) tt-soft α-regular if for every soft α-closed set HE and x∈X such that x⋐̸HE, there exist disjoint soft α-open sets UE and VE such that HE˜⊆UE and x∈VE.
(ⅴ) tt-soft αT3 (resp. tt-soft αT4) if it is both tt-soft α-regular (resp. soft α-normal) and tt-soft αT1.
Remark 3.2. It can be noted that: If FE and GE are disjoint soft set, then x∈FE if and only if x⋐̸GE. This implies that (X,τ,E) is a tt-soft αT2-space if and only if is a soft αT2-space. That is, the concepts of a tt-soft αT2-space and a soft αT2-space are equivalent.
We can say that: (X,τ,E) is tt-soft αT2 if for every x≠y∈X, there exist two disjoint soft α-open sets UE and VE totally contain x and y, respectively.
Remark 3.3. The soft α-regular spaces imply a strict condition on the shape of soft α-open and soft α-closed subsets. To explain this matter, let FE be a soft α-closed set such that x∉HE. Then we have two cases:
(ⅰ) There are e,e′∈E such that x∉H(e) and x∈H(e′). This case is impossible because there do not exist two disjoint soft sets UE and VE containing x and HE, respectively.
(ⅱ) For each e∈E, x∉H(e). This implies that HE must be stable.
As a direct consequence, we infer that every soft α-closed and soft α-open subsets of a soft α-regular space must be stable. However, this matter does not hold on the tt-soft α-regular spaces because we replace a partial non-belong relation by a total non-belong relation. Therefore a tt-soft α-regular space need not be stable.
Proposition 3.4. (ⅰ) Every tt-soft αTi-space is soft αTi for i=0,1,4.
(ⅱ) Every soft α-regular space is tt-soft α-regular.
(ⅲ) Every soft αT3-space is tt-soft αT3.
Proof. The proofs of (ⅰ) and (ⅱ) follow from the fact that a total non-belong relation ⋐̸ implies a partial non-belong relation ∉.
To prove (ⅲ), it suffices to prove that a soft αTi-space is tt-soft αTi when (X,τ,E) is soft α-regular. Suppose x≠y∈X. Then there exist two soft α-open sets UE and VE such that x∈UE and y∉UE; and y∈VE and x∉VE. Since UE and VE are soft α-open subsets of a soft α-regular space, then they are stable. So y⋐̸UE and x⋐̸VE. Thus (X,τ,E) is tt-soft αT1. Hence, we obtain the desired result.
To clarify that the converse of the above proposition does not hold in general, we give the following examples.
Example 3.5. Let E={e1,e2} and τ={˜Φ,˜X,GiE:i=1,2,3} be a soft topology on X={x,y}, where
G1E={(e1,{x}),(e2,X)};G2E={(e1,X),(e2,{y})}andG3E={(e1,{x}),(e2,{y})}. |
One can examine that τ=τα. Then (X,τ,E) is a soft αT1-space. On the other hand, it is not tt-soft αT0 because there does not exist a soft α-open set containing one of the points x or y such that the other point does not totally belong to it.
Example 3.6. Let E={e1,e2} and τ={˜Φ,˜X,GiE:i=1,2,...,8} be a soft topology on X={x,y}, where
G1E={(e1,X),(e2,{x})};G2E={(e1,∅),(e2,{y})};G3E={(e1,{y}),(e2,∅)};G4E={(e1,{y}),(e2,{y})};G5E={(e1,{x}),(e2,{y})};G6E={(e1,X),(e2,{y})};G7E={(e1,{x}),(e2,∅)}andG8E={(e1,X),(e2,∅)}. |
By calculating, we find that τα=τ.
Then (X,τ,E) is a soft αT4-space. On the other hand, there does not exist a soft α-open set totally containing x such that y does not totally belong to it. So (X,τ,E) is not a tt-soft αT1-space, hence it is not tt-soft αT4.
Example 3.7. Let X be any universal set X and E be any set of parameters such that |X|≥2 and |E|≥2. The discrete soft topology (X,τ,E) is a tt-soft α-regular space, but it is not soft α-regular. Hence, it is a tt-soft αT3-space, but it is not soft αT3.
Before we show the relationship between tt-soft αTi-spaces, we need to prove the following useful lemma.
Lemma 3.8. (X,τ,E) is a tt-soft αT1-space if and only if xE is soft α-closed for every x∈X.
Proof. Necessity: For each yi∈X∖{x}, there is a soft α-open set GiE such that yi∈GiE and x⋐̸GiE. Therefore X∖{x}=⋃i∈IGi(e) and x⋐̸⋃i∈IGi(e) for each e∈E. Thus ˜⋃i∈IGiE=~X∖{x} is soft α-open. Hence, xE is soft α-closed.
Sufficiency: Let x≠y. By hypothesis, xE and yE are soft α-closed sets. Then xcE and ycE are soft α-open sets such that x∈(yE)c and y∈(xE)c. Obviously, y⋐̸(yE)c and x⋐̸(xE)c. Hence, (X,τ,E) is tt-soft αT1.
Proposition 3.9. Every tt-soft αTi-space is tt-soft αTi−1 for i=1,2,3,4.
Proof. We prove the proposition in the cases of i=3,4. The other cases follow similar lines.
For i=3, let x≠y in a tt-soft αT3-space (X,τ,E). Then xE is soft α-closed. Since y⋐̸xE and (X,τ,E) is tt-soft α-regular, then there are disjoint soft α-open sets GE and FE such that xE˜⊆GE and y∈FE. Therefore (X,τ,E) is tt-soft αT2.
For i=4, let x∈X and HE be a soft α-closed set such that x⋐̸HE. Since (X,τ,E) is tt-soft αT1, then xE is soft α-closed. Since xE˜⋂HE=˜Φ and (X,τ,E) is soft α-normal, then there are disjoint soft α-open sets GE and FE such that HE˜⊆GE and xE˜⊆FE. Hence, (X,τ,E) is tt-soft αT3.
The following examples show that the converse of the above proposition is not always true.
Example 3.10. Let (X,τ,E) be a soft topological space given in Example (3.6). For x≠y, we have G4E is a soft α-open set such that y∈G4E and x⋐̸G4E. Then (X,τ,E) is tt-soft αT0. However, it is not tt-soft αT1 because there does not exist a soft α-open set totally containing x and does not totally contain y.
Example 3.11. Let E be any set of parameters and τ={˜Φ,GE˜⊆N:GcE is finite} be a soft topology on the set of natural numbers N. It is clear that a soft subset of (N,τ,E) is soft α-open if and only if it is soft open. For each x≠y∈N, we have ~N∖{y} and ~N∖{x} are soft α-open sets such that x∈~N∖{y} and y⋐̸~N∖{y}; and y∈~N∖{x} and x⋐̸~N∖{x}. Therefore (N,τ,E) is tt-soft αT1. On the other hand, there do not exist two disjoint soft α-open sets except for the null and absolute soft sets. Hence, (N,τ,E) is not tt-soft αT2.
Example 3.12. It is well known that a soft topological space is a classical topological space if E is a singleton. Then it suffices to consider examples that satisfy an αT2-space but not αT3; satisfy an αT3-space but not αT4.
In what follows, we establish some properties of tt-soft αTi and tt-soft α-regular.
Lemma 3.13. Let UE be a subset of (X,τ,E) and x∈X. Then x⋐̸¯UEα iff there exists a soft α-open set VE totally containing x such that UE˜⋂VE=˜Φ.
Proof. Let x⋐̸¯UEα. Then x∈(¯UEα)c=VE. So UE˜⋂VE=˜Φ. Conversely, if there exists a soft α-open set VE totally containing x such that UE˜⋂VE=˜Φ, then UE⊆VcE. Therefore ¯UEα⊆VcE. Since x⋐̸VcE, then x⋐̸¯UEα.
Proposition 3.14. If (X,τ,E) is a tt-soft αT0-space, then ¯xEα≠¯yEα for every x≠y∈X.
Proof. Let x≠y in a tt-soft αT0-space. Then there is a soft α-open set UE such that x∈UE and y⋐̸UE or y∈UE and x⋐̸UE. Say, x∈UE and y⋐̸UE. Now, yE˜⋂UE=˜Φ. So, by the above lemma, x⋐̸¯yEα. But x∈¯xEα. Hence, we obtain the desired result.
Corollary 3.15. If (X,τ,E) is a tt-soft αT0-space, then ¯Pxeα≠¯Pye′α for all x≠y and e,e′∈E.
Theorem 3.16. Let E be a finite set. Then (X,τ,E) is a tt-soft αT1-space if and only if xE=˜⋂{UE:x∈UE∈τα} for each x∈X.
Proof. To prove the "if" part, let y∈X. Then for each x∈X∖{y}, we have a soft α-open set UE such that x∈UE and y⋐̸UE. Therefore y⋐̸˜⋂{UE:xE˜⊆UE∈τα}. Since y is chosen arbitrary, then the desired result is proved.
To prove the "only if" part, let the given conditions be satisfied and let x≠y. Let ∣E∣=m. Since y⋐̸xE, then for each j=1,2,...,m there is a soft α-open set UiE such that y∉Ui(ej) and x∈UiE. Therefore ˜⋂mi=1UiE is a soft α-open set such that y⋐̸˜⋂mi=1UiE and x∈˜⋂mi=1UiE. Similarly, we can get a soft α-open set VE such that y∈VE and x⋐̸VE. Thus (X,τ,E) is a tt-soft αT1-space.
Theorem 3.17. If (X,τ,E) is an extended tt-soft αT1-space, then Pxe is soft α-closed for all Pxe∈˜X.
Proof. It follows from Lemma (3.8) that ~X∖{x} is a soft α-open set. Since (X,τ,E) is extended, then a soft set HE, where H(e)=∅ and H(e′)=X for each e′≠e, is a soft α-open set. Therefore ~X∖{x}˜⋃HE is soft α-open. Thus (~X∖{x}˜⋃HE)c=Pxe is soft α-closed.
Corollary 3.18. If (X,τ,E) is an extended tt-soft αT1-space, then the intersection of all soft α-open sets containing UE is exactly UE for each UE˜⊆˜X.
Proof. Let UE be a soft subset of ˜X. Since Pxe is a soft α-closed set for every Pxe∈UcE, then ˜X∖Pxe is a soft α-open set containing UE. Therefore UE=˜⋂{˜X∖Pxe:Pxe∈UcE}, as required.
Theorem 3.19. A finite (X,τ,E) is tt-soft αT2 if and only if it is tt-soft αT1.
Proof. Necessity: It is obtained from Proposition (3.9).
Sufficiency: For each x≠y, we have xE and yE are soft α-closed sets. Since X is finite, then ˜⋃y∈X∖{x}yE and ˜⋃x∈X∖{y}xE are soft α-closed sets. Therefore (˜⋃y∈X∖{x}yE)c=xE and (˜⋃x∈X∖{y}xE)c=yE are disjoint soft α-open sets. Thus (X,τ,E) is a tt-soft αT2-space.
Corollary 3.20. A finite tt-soft αT1-space is soft α-disconnected.
Remark 3.21. In Example (3.11), note that xE is not a soft α-open set for each x∈N. This clarifies that a soft set xE in a tt-soft αT1-space need not be soft α-open if the universal set is infinite.
Theorem 3.22. (X,τ,E) is tt-soft α-regular iff for every soft α-open subset FE of (X,τ,E) totally containing x, there is a soft α-open set VE such that x∈VE˜⊆¯VEα˜⊆FE.
Proof. Let x∈X and FE be a soft α-open set totally containing x. Then FcE is α-soft closed and xE˜⋂FcE=˜Φ. Therefore there are disjoint soft α-open sets UE and VE such that FcE˜⊆UE and x∈VE. Thus VE˜⊆UcE˜⊆FE. Hence, ¯VEα˜⊆UcE˜⊆FE. Conversely, let FcE be a soft α-closed set. Then for each x⋐̸FcE, we have x∈FE. By hypothesis, there is a soft α-open set VE totally containing x such that ¯VEα˜⊆FE. Therefore FcE˜⊆(¯VEα)c and VE˜⋂(¯VEα)c=˜Φ. Thus (X,τ,E) is tt-soft α-regular, as required.
Theorem 3.23. The following properties are equivalent if (X,τ,E) is a tt-soft α-regular space.
(ⅰ) a tt-soft αT2-space.
(ⅱ) a tt-soft αT1-space.
(ⅲ) a tt-soft αT0-space.
Proof. The directions (i)→(ii) and (ii)→(iii) are obvious.
To prove (iii)→(i), let x≠y in a tt-soft αT0-space (X,τ,E). Then there exists a soft α-open set GE such that x∈GE and y⋐̸GE, or y∈GE and x⋐̸GE. Say, x∈GE and y⋐̸GE. Obviously, x⋐̸GcE and y∈GcE. Since (X,τ,E) is tt-soft α-regular, then there exist two disjoint soft α-open sets UE and VE such that x∈UE and y∈GcE˜⊆VE. Hence, (X,τ,E) is tt-soft αT2.
Proposition 3.24. A finite tt-soft αT2-space (X,τ,E) is tt-soft α-regular.
Proof. Let HE be a soft α-closed set and x∈X such that x⋐̸HE. Then x≠y for each y⋐HE. By hypothesis, there are two disjoint soft α-open sets UiE and ViE such that x∈UiE and y∈ViE. Since {y:y∈X} is a finite set, then there is a finite number of soft α-open sets ViE such that HE˜⊆˜⋃mi=1ViE. Now, ˜⋂mi=1UiE is a soft α-open set containing x and [˜⋃mi=1ViE]˜⋂[˜⋂mi=1UiE]=˜Φ. Hence, (X,τ,E) is tt-soft α-regular.
Corollary 3.25. The following properties are equivalent if (X,τ,E) is finite.
(ⅰ) a tt-soft αT3-space.
(ⅱ) a tt-soft αT2-space.
(ⅲ) a tt-soft αT1-space.
Proof. The directions (i)→(ii) and (ii)→(iii) follow from Proposition (3.9).
The direction (iii)→(ii) follows from Theorem (3.19).
The direction (ii)→(i) follows from Proposition (3.24).
Theorem 3.26. The property of being a tt-soft αTi-space (i=0,1,2,3) is a soft open hereditary.
Proof. We prove the theorem in the case of i=3 and the other cases follow similar lines.
Let (Y,τY,E) be a soft open subspace of a tt-soft αT3-space (X,τ,E). To prove that (Y,τY,E) is tt-soft αT1, let x≠y∈Y. Since (X,τ,E) is a tt-soft αT1-space, then there exist two soft α-open sets GE and FE such that x∈GE and y⋐̸GE; and y∈FE and x⋐̸FE. Therefore x∈UE=˜Y˜⋂GE and y∈VE=˜Y˜⋂FE such that y⋐̸UE and x⋐̸VE. It follows from Proposition (2.17), that UE and VE are soft α-open subsets of (Y,τY,E), so that (Y,τY,E) is tt-soft αT1.
To prove that (Y,τY,E) is tt-soft α-regular, let y∈Y and FE be a soft α-closed subset of (Y,τY,E) such that y⋐̸FE. Then FE˜⋃~Yc is a soft α-closed subset of (X,τ,E) such thaty⋐̸FE˜⋃~Yc. Therefore there exist disjoint soft α-open subsets UE and VE of (X,τ,E) such that FE˜⋃~Yc˜⊆UE and y∈VE. Now, UE˜⋂˜Y and VE˜⋂˜Y are disjoint soft α-open subsets of (Y,τY,E) such that FE˜⊆UE˜⋂˜Y and y∈VE˜⋂˜Y. Thus (Y,τY,E) is tt-soft α-regular.
Hence, (Y,τY,E) is tt-soft αT3, as required.
Theorem 3.27. Let (X,τ,E) be extended and i=0,1,2,3,4. Then (X,τ,E) is tt-soft αTi iff (X,τe) is αTi for each e∈E.
Proof. We prove the theorem in the case of i=4 and one can similarly prove the other cases.
Necessity: Let x≠y in X. Then there exist two soft α-open sets UE and VE such that x∈UE and y⋐̸UE; and y∈VE and x⋐̸VE. Obviously, x∈U(e) and y∉U(e); and y∈V(e) and x∉V(e). Since (X,τ,E) is extended, then it follows from Theorem (2.26) that U(e) and V(e) are α-open subsets of (X,τe) for each e∈E. Thus, (X,τe) is an αT1-space. To prove that (X,τe) is α-normal, let Fe and He be two disjoint α-closed subsets of (X,τe). Let FE and HE be two soft sets given by F(e)=Fe, H(e)=He and F(e′)=H(e′)=∅ for each e′≠e. It follows, from Theorem (2.26) that FE and HE are two disjoint soft α-closed subsets of (X,τ,E). By hypothesis, there exist two disjoint soft α-open sets GE and WE such that FE˜⊆GE and HE˜⊆WE. This implies that F(e)=Fe⊆G(e) and H(e)=He⊆W(e). Since (X,τ,E) is extended, then it follows from Theorem (2.26) that G(e) and W(e) are α-open subsets of (X,τe). Thus, (X,τe) is an α-normal space. Hence, it is an αT4-space.
Sufficiency: Let x≠y in X. Then there exists two α-open subsets Ue and Ve of (X,τe) such that x∈Ue and y∉Ue; and y∈Ve and x∉Ve. Let UE and VE be two soft sets given by U(e)=Ue, V(e)=Ve for each e∈E. Since (X,τ,E) is extended, then it follows from Theorem (2.26) that UE and VE are soft α-open subsets of (X,τ,E) such that x∈UE and y⋐̸UE; and y∈VE and x⋐̸VE. Thus, (X,τ,E) is a tt-soft αT1-space. To prove that (X,τ,E) is soft α-normal, let FE and HE be two disjoint soft α-closed subsets of (X,τ,E). Since (X,τ,E) is extended, then it follows from Theorem (2.26) that F(e) and H(e) are two disjoint α-closed subsets of (X,τe). By hypothesis, there exist two disjoint α-open subsets Ge and We of (X,τe) such that F(e)⊆Ge and H(e)⊆We. Let GE and WE be two soft sets given by G(e)=Ge and W(e)=We for each e∈E. Since (X,τ,E) is extended, then it follows from Theorem (2.26) that GE and WE are two disjoint soft α-open subsets of (X,τ,E) such that FE˜⊆GE and HE˜⊆WE. Thus (X,τ,E) is soft α-normal. Hence, it is a tt-soft αT4-space.
In the following examples, we show that a condition of an extended soft topology given in the above theorem is not superfluous.
Example 3.28. Let E={e1,e2} and τ={˜Φ,˜X,G1E,G2E} be a soft topology on X={x,y}, where
G1E={(e1,{x}),(e2,{y})}andG2E={(e1,{y}),(e2,{x})}. |
One can examine that τ=τα. It is clear that (X,τ,E) is not a tt-soft αT0-space. On the other hand, τe1 and τe2 are the discrete topology on X. Hence, the two parametric topological spaces (X,τe1) and (X,τe2) are αT4.
Theorem 3.29. The property of being a tt-soft αTi-space (i=0,1,2) is preserved under a finite product soft spaces.
Proof. We prove the theorem in case of i=2. The other cases follow similar lines.
Let (X1,τ1,E1) and (X2,τ2,E2) be two tt-soft αT2-spaces and let (x1,y1)≠(x2,y2) in X1×X2. Then x1≠x2 or y1≠y2. Without loss of generality, let x1≠x2. Then there exist two disjoint soft α-open subsets GE1 and HE1 of (X1,τ1,E1) such that x1∈GE1 and x2⋐̸GE1; and x2∈HE1 and x1⋐̸HE1. Obviously, GE1×~X2 and HE1×~X2 are two disjoint soft α-open subsets X1×X2 such that (x1,y1)∈GE1×~X2 and (x2,y2)⋐̸GE1×~X2; and (x2,y2)∈HE1×~X2 and (x1,y1)⋐̸HE1×~X2. Hence, X1×X2 is a tt-soft αT2-space.
Theorem 3.30. The property of being a tt-soft αTi-space is an additive property for i=0,1,2,3,4.
Proof. To prove the theorem in the cases of i=2. Let x≠y∈⊕i∈IXi. Then we have the following two cases:
1. There exists i0∈I such that x,y∈Xi0. Since (Xi0,τi0,E) is tt-soft αT2, then there exist two disjoint soft α-open subsets GE and HE of (Xi0,τi0,E) such that x∈GE and y∈HE. It follows from Theorem (2.29), that GE and HE are disjoint soft α-open subsets of (⊕i∈IXi,τ,E).
2. There exist i0≠j0∈I such that x∈Xi0 and y∈Xj0. Now, ~Xi0 and ~Xj0 are soft α-open subsets of (Xi0,τi0,E) and (Xj0,τj0,E), respectively. It follows from Theorem (2.29), that ~Xi0 and ~Xj0 are disjoint soft α-open subsets of (⊕i∈IXi,τ,E).
It follows from the two cases above that (⊕i∈IXi,τ,E) is a tt-soft αT2-space.
The theorem can be proved similarly in the cases of i=0,1.
To prove the theorem in the cases of i=3 and i=4, it suffices to prove the tt-soft α-regularity and soft α-normality, respectively.
First, we prove the tt-soft α-regularity property. Let FE be a soft α-closed subset of (⊕i∈IXi,τ,E) such that x⋐̸FE. It follows from Theorem (2.29) that FE˜⋂~Xi is soft α-closed in (Xi,τi,E) for each i∈I. Since x∈⊕i∈IXi, there is only i0∈I such that x∈Xi0. This implies that there are disjoint soft α-open subsets GE and HE of (Xi0,τi0,E) such that FE˜⋂~Xi0˜⊆GE and y∈HE. Now, GE˜⋃i≠i0~Xi is a soft α-open subset of (⊕i∈IXi,τ,E) containing FE. The disjointness of GE⋃i≠i0Xi and HE ends the proof that (⊕i∈IXi,τ,E) is a tt-soft α-regular space.
Second, we prove the soft α-normality property. Let FE and HE be two disjoint soft α-closed subsets of (⊕i∈IXi,τ,E). It follows from Theorem (2.29) that FE˜⋂~Xi and HE˜⋂~Xi are soft α-closed in (Xi,τi,E) for each i∈I. Since (Xi,τi,E) is soft α-normal for each i∈I, then there there exist two disjoint soft α-open subsets UiE and ViE of (Xi,τi,E) such that FE˜⋂~Xi˜⊆UiE and HE˜⋂~Xi˜⊆ViE. This implies that FE˜⊆˜⋃i∈IUiE, HE˜⊆˜⋃i∈IViE and [˜⋃i∈IUiE]˜⋂[˜⋃i∈IViE]=˜Φ. Hence, (⊕i∈IXi,τ,E) is a soft α-normal space.
In the following we probe the behaviours of tt-soft αTi-spaces under some soft maps.
Definition 3.31. A map fφ:(X,τ,A)→(Y,θ,B) is said to be:
1. soft α⋆-continuous if the inverse image of soft α-open set is soft α-open.
2. soft α⋆-open (resp. soft α⋆-closed) if the image of soft α-open (resp. soft α-closed) set is soft α-open (resp. soft α-closed).
3. soft α⋆-homeomorphism if it is bijective, soft α⋆-continuous and soft α⋆-open.
Proposition 3.32. Let fφ:(X,τ,A)→(Y,θ,B) be a soft α-continuous map such that f is injective. Then if (Y,θ,B) is a p-soft Ti-space, then (X,τ,A) is a tt-soft αTi-space for i=0,1,2.
Proof. We only prove the proposition for i=2.
Let fφ:(X,τ,A)→(Y,θ,B) be a soft α-continuous map and a≠b∈X. Since f is injective, then there are two distinct points x and y in Y such that f(a)=x and f(b)=y. Since (Y,θ,B) is a p-soft T2-space, then there are two disjoint soft open sets GB and FB such that x∈GB and y∈FB. Now, f−1φ(GB) and f−1φ(FB) are two disjoint soft α-open subsets of (X,τ,A) such that a∈f−1φ(GB) and b∈f−1φ(FB). Thus (X,τ,A) is a tt-soft αT2-space.
In a similar way, one can prove the following result.
Proposition 3.33. Let fφ:(X,τ,A)→(Y,θ,B) be a soft α⋆-continuous map such that f is injective. Then if (Y,θ,B) is a tt-soft αTi-space, then (X,τ,A) is a tt-soft αTi-space for i=0,1,2.
Proposition 3.34. Let fφ:(X,τ,A)→(Y,θ,B) be a bijective soft α-open map. Then if (X,τ,A) is a p-soft Ti-space, then (Y,θ,B) is a tt-soft αTi-space for i=0,1,2.
Proof. We only prove the proposition for i=2.
Let fφ:(X,τ,A)→(Y,θ,B) be a soft α-open map and x≠y∈Y. Since f is bijective, then there are two distinct points a and b in X such that a=f−1(x) and b=f−1(y). Since (X,τ,A) is a p-soft T2-space, then there are two disjoint soft open sets UA and VA such that x∈UA and y∈VA. Now, fφ(UA) and fφ(VA) are two disjoint soft α-open subsets of (Y,θ,B) such that x∈fφ(UA) and y∈fφ(VA). Thus (Y,θ,B) is a tt-soft αT2-space.
In a similar way, one can prove the following result.
Proposition 3.35. Let fφ:(X,τ,A)→(Y,θ,B) be a bijective soft α⋆-open map. Then if (X,τ,A) is a tt-soft αTi-space, then (Y,θ,B) is a tt-soft αTi-space for i=0,1,2.
Proposition 3.36. The property of being tt-soft αTi(i=0,1,2,3,4) is preserved under a soft α⋆-homeomorphism map.
We complete this section by discussing some interrelations between tt-soft αTi-spaces (i=2,3,4) and soft α-compact spaces.
Proposition 3.37. A stable soft α-compact subset of a tt-soft αT2-space is soft α-closed.
Proof. It follows from Proposition (2.21) and Remark (3.2).
Theorem 3.38. Let HE be a soft α-compact subset of a tt-soft αT2-space. If x⋐̸HE, then there are disjoint soft α-open sets UE and VE such that x∈UE and HE⊆VE.
Proof. Let x⋐̸HE. Then x≠y for each y⋐HE. Since (X,τ,E) is a tt-soft αT2-space, then there exist disjoint soft α-open sets UiE and ViE such that x∈UiE and y∈ViE. Therefore {ViE} forms a soft α-open cover of HE. Since HE is soft α-compact, then HE⊆˜⋃i=ni=1ViE. By letting ˜⋃i=ni=1ViE=VE and ˜⋂i=ni=1UiE=UE, we obtain the desired result.
Theorem 3.39. Every soft α-compact and tt-soft αT2-space is tt-soft α-regular.
Proof. Let HE be a soft α-closed subset of soft α-compact and tt-soft αT2-space (X,τ,E) such that x⋐̸HE. Then HE is soft α-compact. By Theorem (3.38), there exist disjoint soft α-open sets UE and VE such that x∈UE and HE⊆VE. Thus, (X,τ,E) is tt-soft α-regular.
Corollary 3.40. Every soft α-compact and tt-soft αT2-space is tt-soft αT3.
Lemma 3.41. Let FE be a soft α-open subset of a soft α-regular space. Then for each Pxe∈FE, there exists a soft α-open set GE such that Pxe∈¯GEα˜⊆FE.
Proof. Let FE be a soft α-open set such that Pxe∈FE. Then x∉FcE. Since (X,τ,E) is soft α-regular, then there exist two disjoint soft α-open sets GE and WE containing x and FcE, respectively. Thus x∈GE˜⊆WcE˜⊆FE. Hence, Pxe∈GE˜⊆¯GEα˜⊆WcE˜⊆FE.
Theorem 3.42. Let HE be a soft α-compact subset of a soft α-regular space and FE be a soft α-open set containing HE. Then there exists a soft α-open set GE such that HE˜⊆GE˜⊆¯GEα˜⊆FE.
Proof. Let the given conditions be satisfied. Then for each Pxe∈HE, we have Pxe∈FE. Therefore there is a soft α-open set WxeE such that Pxe∈WxeE˜⊆¯WxeEα˜⊆FE. Now, {WxeE:Pxe∈FE} is a soft α-open cover of HE. Since HE is soft α-compact, then HE˜⊆˜⋃i=ni=1WxeE. Putting GE=˜⋃i=ni=1WxeE. Thus HE˜⊆GE˜⊆¯GEα˜⊆FE.
Corollary 3.43. If (X,τ,E) is soft α-compact and soft αT3, then it is tt-soft αT4.
Proof. Suppose that F1E and F2E are two disjoint soft α-closed sets. Then F2E˜⊆Fc1E. Since (X,τ,E) is soft α-compact, then F2E is soft α-compact and since (X,τ,E) is soft α-regular, then there is a soft α-open set GE such that F2E˜⊆GE˜⊆¯GEα˜⊆Fc1E. Obviously, F2E˜⊆GE,F1E˜⊆(¯GEα)c and GE˜⋂(¯GEα)c=˜Φ. Thus (X,τ,E) is soft α-normal. Since (X,τ,E) is soft αT3, then it is tt-soft αT1. Hence, it is tt-soft αT4.
In this section, we investigate main features of an α-fixed soft point, in particular, those are related to parametric topological spaces.
Theorem 4.1. Let {Bn:n∈N} be a collection of soft subsets of a soft α-compact space (X,τ,E) satisfying:
(ⅰ) Bn≠˜Φ for each n∈N;
(ⅱ) Bn is a soft α-closed set for each n∈N;
(ⅲ) Bn+1˜⊆Bn for each n∈N.
Then ˜⋂n∈NBn≠˜Φ.
Proof. Suppose that ˜⋂n∈NBn=˜Φ. Then ˜⋃n∈NBcn=˜X. It follows from (ii) that {Bcn:n∈N} is a soft α-open cover of ˜X. By hypothesis of soft α-compactness, there exist i1,i2,...,ik∈N, i1<i2<...<ik such that ˜X=Bci1˜⋃Bci2˜⋃...˜⋃Bcik. It follows from (iii) that Bik˜⊆˜X=Bci1˜⋃Bci2˜⋃...˜⋃Bcik=[Bi1˜⋂Bi2˜⋂...˜⋂Bik]c=Bcik. This yields a contradiction. Thus we obtain the proof that ˜⋂n∈NBn≠˜Φ.
To illustrate the above theorem, we give the following example
Example 4.2. As we mentioned that a soft topological space is a classical topological space if E={e} is a singleton. Then we show the above theorem in the crisp setting. Let τ=R⋃{G⊆R:1∉R} be a (soft) topology on R (it is called an excluding point topology). One can examined that (R,τ,E) is a soft α-compact space. Let {Mn:n∈N} be a collection of soft subsets of (R,τ,E) defined as follows: Mn=N∖{2,...,n+1}; that is M1=N∖{2}, M2=N∖{2,3}, and so on. It is clear that Mn satisfied the three conditions (i)-(iii) given in the above theorem. Now, 1∈⋂n∈NMn, as required.
Proposition 4.3. Let (X,τ,E) be a soft α-compact and soft αT′2-space and gφ:(X,τ,E)→(X,τ,E) be a soft α⋆-continuous map. Then there exists a unique soft point Pxe∈˜X of gφ.
Proof. Let {B1=gφ(˜X) and Bn=gφ(Bn−1)=gnφ(˜X) for each n∈N} be a family of soft subsets of (X,τ,E). It is clear that Bn+1˜⊆Bn for each n∈N. Since gφ is soft α⋆-continuous, then Bn is a soft α-compact set for each n∈N and since (X,τ,E) is soft αT′2, then Bn is also a soft α-closed set for each n∈N. It follows from Theorem (4.1) that (H,E)=˜⋂n∈NBn is a non null soft set. Note that gφ(H,E)=gφ(˜⋂n∈Ngnφ(˜X))˜⊆˜⋂n∈Ngn+1φ(˜X)˜⊆˜⋂n∈Ngnφ(˜X)=(H,E). To show that (H,E)˜⊆gφ(H,E), suppose that there is a Pxe∈(H,E) such that Pxe∉gφ(H,E). Let Cn=g−1φ(Pxe)˜⋂Bn. Obviously, Cn≠˜Φ and Cn˜⊆Cn−1 for each n∈N. By Theorem (2.15), Cn is a soft α-closed set for each n∈N; and by Theorem (4.1), there exists a soft point Pym such that Pym∈g−1φ(Pxe)˜⋂Bn. Therefore Pxe=gφ(Pym)∈gφ(H,E). This is a contradiction. Thus, gφ(H,E)=(H,E). Hence, the proof is complete.
Definition 4.4. (ⅰ) (X,τ,E) is said to have an α-fixed soft point property if every soft α⋆-continuous map gφ:(X,τ,E)→(X,τ,E) has a fixed soft point.
(ⅱ) A property is said to be an α⋆-soft topological property if the property is preserved by soft α⋆-homeomorphism maps.
Proposition 4.5. The property of being an α-fixed soft point is an α⋆-soft topological property.
Proof. Let (X,τ,E) and (Y,θ,E) be a soft α⋆-homeomorphic. Then there is a bijective soft map fφ:(X,τ,E)→(Y,θ,E) such that fφ and f−1φ are soft α⋆-continuous. Since (X,τ,E) has an α-fixed soft point property, then every soft α⋆-continuous map gφ:(X,τ,E)→(X,τ,E) has an α-fixed soft point. Now, let hφ:(Y,θ,E)→(Y,θ,E) be a soft α⋆-continuous. Obviously, hφ∘fφ:(X,τ,E)→(Y,θ,E) is a soft α⋆-continuous. Also, f−1φ∘hφ∘fφ:(X,τ,E)→(X,τ,E) is a soft α⋆-continuous. Since (X,τ,E) has an α-fixed soft point property, then f−1φ(hφ(fφ(Pxe)))=Pxe for some Pxe∈˜X. consequently, fφ(f−1φ(hφ(fφ(Pxe))))=fφ(Pxe). This implies that hφ(fφ(Pxe))=fφ(Pxe). Thus fφ(Pxe) is an α-fixed soft point of hφ. Hence, (Y,θ,E) has an α-fixed soft point property, as required.
Before we investigate a relationship between soft topological space and their parametric topological spaces in terms of possessing a fixed (soft) point, we need to prove the following result.
Theorem 4.6. Let τ be an extended soft topology on X. Then a soft map gφ:(X,τ,E)→(Y,θ,E) is soft α⋆-continuous if and only if a map g:(X,τe)→(Y,θϕ(e)) is α⋆-continuous.
Proof. Necessity: Let U be an α-open subset of (Y,θϕ(e)). Then there exists a soft α-open subset GE of (Y,θ,E) such that G(ϕ(e))=U. Since gφ is a soft α⋆-continuous map, then g−1ϕ(GE) is a soft α-open set. From Definition (2.9), it follows that a soft subset g−1ϕ(GE)=(g−1ϕ(G))E of (X,τ,E) is given by g−1ϕ(G)(e)=g−1(G(ϕ(e))) for each e∈E. By hypothesis, τ is an extended soft topology on X, we obtain from Theorem (2.26) that a subset g−1(G(ϕ(e)))=g−1(U) of (X,τe) is α-open. Hence, a map g is α⋆-continuous.
Sufficiency: Let GE be a soft α-open subset of (Y,θ,E). Then from Definition (2.9), it follows that a soft subset g−1ϕ(GE)=(g−1ϕ(G))E of (X,τ,E) is given by g−1ϕ(G)(e)=g−1(G(ϕ(e))) for each e∈E. Since a map g is α⋆-continuous, then a subset g−1(G(ϕ(e))) of (X,τe) is α-open. By hypothesis, τ is an extended soft topology on X, we obtain from Theorem (2.26) that g−1ϕ(GE) is a soft α-open subset of (X,τ,E). Hence, a soft map gφ is soft α⋆-continuous.
Definition 4.7. (X,τ) is said to have an α-fixed point property if every α⋆-continuous map g:(X,τ)→(X,τ) has a fixed point.
Proposition 4.8. (X,τ,E) has the property of an α-fixed soft point iff (X,τe) has the property of an α-fixed point for each e∈E.
Proof. Necessity: Let (X,τ,E) has the property of an α-fixed soft point. Then every soft α⋆-continuous map gφ:(X,τ,E)→(X,τ,E) has a fixed soft point. Say, Pxe. It follows from the above theorem that ge:(X,τe)→(X,θϕ(e)) is α⋆-continuous. Since Pxe is a fixed soft point of gφ, then it must be that ge(x)=x. Thus, ge has a fixed point. Hence, we obtain the desired result.
Sufficiency: Let (X,τe) has the property of an α-fixed point for each e∈E. Then every α⋆-continuous map ge:(X,τe)→(X,θϕ(e)) has a fixed point. Say, x. It follows from the above theorem that gφ:(X,τ,E)→(X,θ,E) is soft α⋆-continuous. Since x is a fixed point of ge, then it must be that gφ(Pxe)=Pxe. Thus, gφ has a fixed soft point. Hence, we obtain the desired result.
This work presents new types of soft separation axioms with respect to three factors:
(ⅰ) ordinary points.
(ⅱ) total belong and total non-belong relations.
(ⅲ) soft α-open sets.
We show the interrelationships between these soft separation axioms and investigate some properties. The main contributions of this work are the following:
(ⅰ) formulate new soft separation axioms, namely tt-soft αTi(i=0,1,2,3,4) and tt-soft α-regular spaces.
(ⅱ) illustrate the relationships between them as well as with soft αTi(i=0,1,2,3,4) and soft α-regular spaces.
(ⅲ) study the "transmission" of these soft separation axioms between soft topological space and its parametric topological spaces.
(ⅳ) give some conditions that guarantee the equivalence of tt-soft αTi(i=0,1,2) and the equivalence of tt-soft αTi(i=1,2,3).
(ⅴ) characterize some of these soft separation axioms such as tt-soft αT1 and tt-soft α-regular spaces
(ⅵ) explore the interrelations of some of these soft separation axioms and soft compact spaces.
(ⅶ) discuss the behaviours of these soft separation axioms with some notions such as product soft spaces and sum of soft topological spaces.
(ⅷ) define α-fixed soft point and establish fundamental properties.
Soft separation axioms are among the most widespread and important concepts in soft topology because they are utilized to classify the objects of study and to construct different families of soft topological spaces. In this work, we have introduced new soft separation axioms with respect to ordinary points by using total belong and total non-belong relations. This way of definition helps us to generalize existing comparable properties via general topology and to remove a strict condition of the shape of soft open and closed subsets of soft α-regular spaces. In general, we study their main properties and illustrate the interrelations between them and some soft topological notions such as soft compactness, product soft spaces and sum of soft topological spaces. We complete this work by defining α-fixed soft point theorem and investigating its basic properties.
We plan in the upcoming works to study the concepts and results presented herein by using some celebrated types of generalizations of soft open sets such as soft preopen, soft b-open and soft β-open sets. In addition, we will explore these concepts on some contents such as supra soft topology and fuzzy soft topology. In the end, we hope that the concepts initiated herein will find their applications in many fields soon.
The authors declare that they have no competing interests.
The authors would like to thank the referees for their valuable comments which help us to improve the manuscript.
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