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Connectedness Generalizations Using the Concept of Adherence Dominators

Received: 6 March 2022     Accepted: 30 March 2022     Published: 23 April 2022
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Abstract

An adherence dominator on a topological space X is a function π from the collection of filterbases on X to the collection of closed subsets of X satisfying Ω ⊂ πΩ where Ω is the adherence of Ω and , where ∑F represents the collection of open sets containing F. The π-adherence may be adherence, θ-adherence, u-adherence s-adherence, f-adherence, δ-adherence, etc., of a filterbase. Pervin deined a partition (or a separation) of a set in a topological space as a pair of subsets (P,Q) satisfying P∩clQ = clP∩Q = ∅, where clP represents the closure of P and a set K is said to be connected if K = ∅ or K ≠ PQ where (P,Q) is a partition. In this paper, a πpartition (or a πseparation) is a pair of subsets (P,Q) satisfying PπQ = πPQ = ∅ where π is an adherence dominator and a subset K of a space X is πconnected relative to X lf K = ∅ or there is no πpartition (P,Q) such that K = PQ. This paper investigates these new forms of connectedness. Theorems due to A. D. Wallace and J. D. Kline are generalized. Geralizations of C-compact spaces and functionally compact spaces are also presented.

Published in American Journal of Applied Mathematics (Volume 10, Issue 2)
DOI 10.11648/j.ajam.20221002.13
Page(s) 43-50
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2022. Published by Science Publishing Group

Keywords

Flters, Adherence Dominators, Connectedness, πseparation, πclosed

References
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[2] J. P. Clay and J. E. Joseph, On a connectivity propertty induced by the θ-closure operator, Illinois Journal of Mathematics Vol. 25 (1981), 267-278.
[3] T. A. Edwards, J. E Joseph, M. H. Kwack and B. M. P. Nayar, Compactness via θ-closed and θ-rigid subsets, Journal of Advanced studies in Topology, Vol. 5 (3) (2014), 28-34.
[4] T. A. Edwards, J. E Joseph, M. H. Kwack and B. M. P. Nayar, Compactness via adherence dominators, Journal of Advanced studies in Topology, Vol. 5 (4) (2014), 8-15.
[5] M. S. Espelie and J. E. Joseph, Some properties of θ- closure, Can. J. Math. Vol. 33, No. 1, 1981, 142-149.
[6] M. S. Espelie, J. E. Joseph and M. H. Kwack, Applications of the u-closure operator, Proc. Amer. Math. Soc. Vol. 83 (1981), 167-174.
[7] L. L. Herrington, Characterizations of Urysohn-closed spaces, Proc. Amer. Math. Soc. Vol. 53 (1976), 435- 439.
[8] L. L. Herrington, Characterizations of ompletely Hausdorff-closed spaces, Proc. Amer. Math. Soc. Vol. 55 (1976), 140-144.
[9] L.L.Herrington, Characterizations of regular-closed spaces, Math. Chronicle Vol. 5 (1977), 168 -178.
[10] J. E. Joseph, Urysohn-closed and minimal Urysohn spaces, Proc. Amer. Math. Soc. Vol. 68 (2) (1978), 235-242.
[11] J. E. Joseph, Regular-closed and minimal regular spaces, Canad. Math. Bull. Vol. 22 (4), (1979), 491 -497.
[12] J. E. Joseph, P-closed and minimal p-spaces from adherence dominators and graphs, Rev. Roumaine Math. Pures Appl. Vol. 25 No.7 (1980), 1047-1057.
[13] J. E. Joseph, M. H. Kwack and B. M. P. Nayar, Weak continuity forms, graph conditions and applications, Scientae Mathematicae Vol. 2 No.1 (1999), 65-88.
[14] J. E. Joseph and B. M. P. Nayar, A study of topological properties via adherence dominators, Lecture Notes in Nonlinear Analysis, Juliusz Shauder Center for Nonlinear Studies, (A monograph accepted for publication)
[15] J. L. Kelley, General Topology, Van Nostrand Reinhold Company, New York (1969).
[16] J. R. Kline, A theorem concerning connected point sets, Fundamenta Math. Vol 3 (1922), 238-239.
[17] B. Knaster and K. Kuratowski, Sur les ensembles connexes, Fundemanta Math. Vol 2 (1921) 206-255.
[18] N. Levine, Extensions of topologies, Amer. Math. Monthly Vol. 71, (1964), 22-25.
[19] W. J. Pervin, Foundations of Genral Topology, Academic Press, New York (1964).
[20] C. T. Scarborough, Minimal Urysohn spaces, Pac. Journal of Math. Vol. 27 (1968), 611-617.
[21] M. K. Singal and A. Mathur, On nearly compact spaces, Boll. Un. Mat. Ital. Vol. (4) 2 (1969), 702-710.
[22] N. V. Veli˘ cko, H-closed topological spaces,Math.Sb (N.S) 70 (112) (1966), 98 - 112 (Russian), (Amer. Math Soc. Transl. 78(2) (1968), 103-118).
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  • APA Style

    James Edward Joseph, Bhamini M. P. Nayar. (2022). Connectedness Generalizations Using the Concept of Adherence Dominators. American Journal of Applied Mathematics, 10(2), 43-50. https://doi.org/10.11648/j.ajam.20221002.13

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    ACS Style

    James Edward Joseph; Bhamini M. P. Nayar. Connectedness Generalizations Using the Concept of Adherence Dominators. Am. J. Appl. Math. 2022, 10(2), 43-50. doi: 10.11648/j.ajam.20221002.13

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    AMA Style

    James Edward Joseph, Bhamini M. P. Nayar. Connectedness Generalizations Using the Concept of Adherence Dominators. Am J Appl Math. 2022;10(2):43-50. doi: 10.11648/j.ajam.20221002.13

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  • @article{10.11648/j.ajam.20221002.13,
      author = {James Edward Joseph and Bhamini M. P. Nayar},
      title = {Connectedness Generalizations Using the Concept of Adherence Dominators},
      journal = {American Journal of Applied Mathematics},
      volume = {10},
      number = {2},
      pages = {43-50},
      doi = {10.11648/j.ajam.20221002.13},
      url = {https://doi.org/10.11648/j.ajam.20221002.13},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20221002.13},
      abstract = {An adherence dominator on a topological space X is a function π from the collection of filterbases on X to the collection of closed subsets of X satisfying Ω ⊂ πΩ where Ω is the adherence of Ω and , where ∑F represents the collection of open sets containing F. The π-adherence may be adherence, θ-adherence, u-adherence s-adherence, f-adherence, δ-adherence, etc., of a filterbase. Pervin deined a partition (or a separation) of a set in a topological space as a pair of subsets (P,Q) satisfying P∩clQ = clP∩Q = ∅, where clP represents the closure of P and a set K is said to be connected if K = ∅ or K ≠ PQ where (P,Q) is a partition. In this paper, a πpartition (or a πseparation) is a pair of subsets (P,Q) satisfying P∩πQ = πP∩Q = ∅ where π is an adherence dominator and a subset K of a space X is πconnected relative to X lf K = ∅ or there is no πpartition (P,Q) such that K = PQ. This paper investigates these new forms of connectedness. Theorems due to A. D. Wallace and J. D. Kline are generalized. Geralizations of C-compact spaces and functionally compact spaces are also presented.},
     year = {2022}
    }
    

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    T1  - Connectedness Generalizations Using the Concept of Adherence Dominators
    AU  - James Edward Joseph
    AU  - Bhamini M. P. Nayar
    Y1  - 2022/04/23
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    N1  - https://doi.org/10.11648/j.ajam.20221002.13
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    T2  - American Journal of Applied Mathematics
    JF  - American Journal of Applied Mathematics
    JO  - American Journal of Applied Mathematics
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    EP  - 50
    PB  - Science Publishing Group
    SN  - 2330-006X
    UR  - https://doi.org/10.11648/j.ajam.20221002.13
    AB  - An adherence dominator on a topological space X is a function π from the collection of filterbases on X to the collection of closed subsets of X satisfying Ω ⊂ πΩ where Ω is the adherence of Ω and , where ∑F represents the collection of open sets containing F. The π-adherence may be adherence, θ-adherence, u-adherence s-adherence, f-adherence, δ-adherence, etc., of a filterbase. Pervin deined a partition (or a separation) of a set in a topological space as a pair of subsets (P,Q) satisfying P∩clQ = clP∩Q = ∅, where clP represents the closure of P and a set K is said to be connected if K = ∅ or K ≠ PQ where (P,Q) is a partition. In this paper, a πpartition (or a πseparation) is a pair of subsets (P,Q) satisfying P∩πQ = πP∩Q = ∅ where π is an adherence dominator and a subset K of a space X is πconnected relative to X lf K = ∅ or there is no πpartition (P,Q) such that K = PQ. This paper investigates these new forms of connectedness. Theorems due to A. D. Wallace and J. D. Kline are generalized. Geralizations of C-compact spaces and functionally compact spaces are also presented.
    VL  - 10
    IS  - 2
    ER  - 

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Author Information
  • Retired, Department of Mathematics, Howard University, Washington, USA

  • Department of Mathematics, Morgan State University, Baltimore, USA

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