Research article

On some types of functions and a form of compactness via $ \omega _{s} $-open sets

  • Received: 03 June 2021 Accepted: 02 November 2021 Published: 10 November 2021
  • MSC : 54A10, 54A20, 54C08, 54C10

  • 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 irresoluteness. Also, $ \omega _{s} $-openness which lies strictly between openness and semi-openness is defined. Sufficient conditions for the equivalence between $ \omega _{s} $-openness and openness, and between $ \omega _{s} $-openness and semi-openness are given. Moreover, pre-$ \omega _{s} $ -openness which is a strong form of $ \omega _{s} $-openness and independent of each of openness and pre-semi-openness is introduced. Furthermore, slight $ \omega _{s} $-continuity as a new class of functions which lies between slight continuity and slight semi-continuity is introduced. Several results related to slight $ \omega _{s} $-continuity are introduced, in particular, sufficient conditions for the equivalence between slight $ \omega _{s} $ -continuity and slight continuity, and between slight $ \omega _{s} $ -continuity and slight semi-continuity are given. In addition to these, $ \omega _{s} $-compactness as a new class of topological spaces that lies strictly between compactness and semi-compactness is introduced. It is proved that locally countable compact topological spaces are $ \omega _{s} $ -compact. Also, it is proved that anti-locally countable $ \omega _{s} $ -compact topological spaces are semi-compact. Several implications, examples, counter-examples, characterizations, and mapping theorems are introduced related to the above concepts are introduced.

    Citation: Samer Al Ghour. On some types of functions and a form of compactness via $ \omega _{s} $-open sets[J]. AIMS Mathematics, 2022, 7(2): 2220-2236. doi: 10.3934/math.2022126

    Related Papers:

  • 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 irresoluteness. Also, $ \omega _{s} $-openness which lies strictly between openness and semi-openness is defined. Sufficient conditions for the equivalence between $ \omega _{s} $-openness and openness, and between $ \omega _{s} $-openness and semi-openness are given. Moreover, pre-$ \omega _{s} $ -openness which is a strong form of $ \omega _{s} $-openness and independent of each of openness and pre-semi-openness is introduced. Furthermore, slight $ \omega _{s} $-continuity as a new class of functions which lies between slight continuity and slight semi-continuity is introduced. Several results related to slight $ \omega _{s} $-continuity are introduced, in particular, sufficient conditions for the equivalence between slight $ \omega _{s} $ -continuity and slight continuity, and between slight $ \omega _{s} $ -continuity and slight semi-continuity are given. In addition to these, $ \omega _{s} $-compactness as a new class of topological spaces that lies strictly between compactness and semi-compactness is introduced. It is proved that locally countable compact topological spaces are $ \omega _{s} $ -compact. Also, it is proved that anti-locally countable $ \omega _{s} $ -compact topological spaces are semi-compact. Several implications, examples, counter-examples, characterizations, and mapping theorems are introduced related to the above concepts are introduced.



    加载中


    [1] H. Hdeib, $\omega $-closed mappings, Revista Colomb. de Mat., 16 (1982), 65–78.
    [2] A. Al-Omari, H. Al-Saadi, On $\omega ^{\ast }$-connected spaces, Songklanakarin J. Sci. Technol., 42 (2020), 280–283. doi: 10.14456/sjst-psu.2020.36. doi: 10.14456/sjst-psu.2020.36
    [3] S. Al Ghour, B. Irshidat, On $\theta _{\omega }$ continuity, Heliyon, 6 (2020), e03349. doi: 10.1016/j.heliyon.2020.e03349. doi: 10.1016/j.heliyon.2020.e03349
    [4] L. L. L. Butanas, M. A. Labendia, $\theta _{\omega }$-connected space and $\theta _{\omega }$-continuity in the product space, Poincare J. Anal. Appl., 7 (2020), 79–88.
    [5] R. M. Latif, Theta-$\omega $-mappings in topological spaces, WSEAS Trans. Math., 19 (2020), 186–207. doi: 10.37394/23206.2020.19.18. doi: 10.37394/23206.2020.19.18
    [6] N. Noble, Some thoughts on countable Lindelöf product, Topol. Appl., 259 (2019), 287–310. doi: 10.1016/j.topol.2019.02.037. doi: 10.1016/j.topol.2019.02.037
    [7] H. H. Al-Jarrah, A. Al-Rawshdeh, E. M. Al-Saleh, K. Y. Al-Zoubi, Characterization of $R\omega O(X)$ sets by using $\delta _{\omega }$-cluster points, Novi Sad J. Math., 49 (2019), 109–122. doi: 10.30755/NSJOM.08786. doi: 10.30755/NSJOM.08786
    [8] C. Carpintero, N. Rajesh, E. Rosas, On real valued $\omega $ -continuous functions, Acta Univ. Sapientiae Math., 10 (2018), 242–248. doi: 10.2478/ausm-2018-0019. doi: 10.2478/ausm-2018-0019
    [9] S. H. Al Ghour, Three new weaker notions of fuzzy open sets and related covering concepts, Kuwait J. Sci., 4 (2017), 48–57.
    [10] S. H. Al Ghour, On several types of continuity and irresoluteness in $L$-topological spaces, Kuwait J. Sci., 45 (2018), 9–14.
    [11] S. Al Ghour, W. Hamed, On two classes of soft sets in soft topological spaces, Symmetry, 12 (2020), 265. doi: 10.3390/sym12020265. doi: 10.3390/sym12020265
    [12] N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly, 70 (1963), 36–41. doi: 10.1080/00029890.1963.11990039. doi: 10.1080/00029890.1963.11990039
    [13] N. Biswas, On some mappings in topological spaces, Bull. Calcutta Math. Soc., 61 (1969), 127–135.
    [14] S. G. Crossley, S. K. Hildebrand, Semi-topological properties, Fund. Math., 74 (1972), 233–254.
    [15] T. M. Nour, Slightly semi-continuous functions, Bull. Calcutta Math. Soc., 87 (1995), 187–190.
    [16] J. A. Hassan, M. A. Labendia, $\theta _{s}$-open sets and $ \theta _{s}$-continuity of maps in the product space, J. Math. Comput. Sci., 25 (2022), 182–190. doi: 10.22436/jmcs.025.02.07. doi: 10.22436/jmcs.025.02.07
    [17] J. D. Cao, A. McCluskey, Topological transitivity in quasi-continuous dynamical systems, Topol. Appl., 301 (2021), 107496. doi: 10.1016/j.topol.2020.107496. doi: 10.1016/j.topol.2020.107496
    [18] C. Granados, New results on semi-i-convergence, T. A. Razmadze Math. In., 175 (2021), 199–204.
    [19] S. Kowalczyk, M. Turowska, On continuity in generalized topology, Topol. Appl., 297 (2021), 107702. doi: 10.1016/j.topol.2021.107702. doi: 10.1016/j.topol.2021.107702
    [20] G. Ivanova, E. Wagner-Bojakowska, $A$-continuity and measure, Lith. Math. J., 61 (2021), 239–245. doi: 10.1007/s10986-021-09514-z. doi: 10.1007/s10986-021-09514-z
    [21] P. Szyszkowska, Separating sets by functions and by sets, Topol. Appl., 284 (2020), 107404. doi: 10.1016/j.topol.2020.107404. doi: 10.1016/j.topol.2020.107404
    [22] S. Sharma, S. Billawria, T. Landol, On almost $\alpha $ -topological vector spaces, Missouri J. Math. Sci., 32 (2020), 80–87. doi: 10.35834/2020/3201080. doi: 10.35834/2020/3201080
    [23] S. E. Han, Semi-separation axioms of the infinite Khalimsky topological sphere, Topol. Appl., 275 (2020), 107006. doi: 10.1016/j.topol.2019.107006. doi: 10.1016/j.topol.2019.107006
    [24] E. Przemska, The lattices of families of regular sets in topological spaces, Math. Slovaca, 70 (2020), 477–488. doi: 10.1515/ms-2017-0365. doi: 10.1515/ms-2017-0365
    [25] O. V. Maslyuchenko, D. P. Onypa, A quasi-locally constant function with given cluster sets, Eur. J. Math., 6 (2020), 72–79. doi: 10.1007/s40879-020-00397-x. doi: 10.1007/s40879-020-00397-x
    [26] J. Sanabria, E. Rosas, L. Vasquez, On inversely $\theta $ -semi-open and inversely $\theta $-semi-closed functions, Mate. Studii, 53 (2020), 92–99. doi: 10.30970/ms.53.1.92-99. doi: 10.30970/ms.53.1.92-99
    [27] A. S. Salama, Sequences of topological near open and near closed sets with rough applications, Filomat, 1 (2020), 51–58. doi: 10.2298/FIL2001051S. doi: 10.2298/FIL2001051S
    [28] A. A. Azzam, A. A. Nasef, Some topological notations via Maki's $\Lambda $-sets, Complexity, 2020 (2020), 4237462. doi: 10.1155/2020/4237462. doi: 10.1155/2020/4237462
    [29] S. Al Ghour, K. Mansur, Between open sets and semi-open sets, Univ. Sci., 23 (2018), 9–20. doi: 10.11144/Javeriana.SC23-1.bosa. doi: 10.11144/Javeriana.SC23-1.bosa
    [30] M. Gillman, M. Jerison, Rings of continuous functions, D Van Nostrant Company Incorporated, 1960.
    [31] K. Al-Zoubi, B. Al-Nashef, The topology of $\omega $-open subsets, Al-Manarah J., 9 (2003), 169–179.
    [32] H. Z. Hdeib, $\omega $-continuous functions, Dirasat J., 16 (1989), 136–142.
    [33] S. Al Ghour, Certain covering properties related to paracompactness, University of Jordan, Amman, Jordan, 1999.
    [34] A. R. Singal, R. C. Jain, Slightly continuous mappings, J. Indian Math. Soc., 64 (1997), 195–203.
    [35] C. Dorsett, semi-compactness, semi separation axioms, and product spaces, Bull. Malaysian Math. Soc., 2 (1981), 21–28.
  • Reader Comments
  • © 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1593) PDF downloads(77) Cited by(0)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog