Connectedness Generalizations Using the Concept of Adherence Dominators

James Edward Joseph; Bhamini M. P. Nayar

doi:doi:10.11648/j.ajam.20221002.13

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Connectedness Generalizations Using the Concept of Adherence Dominators

James Edward Joseph, Bhamini M. P. Nayar

Published in American Journal of Applied Mathematics (Volume 10, Issue 2)

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 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.

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.
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[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.
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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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TY  - JOUR
T1  - Connectedness Generalizations Using the Concept of Adherence Dominators
AU  - James Edward Joseph
AU  - Bhamini M. P. Nayar
Y1  - 2022/04/23
PY  - 2022
N1  - https://doi.org/10.11648/j.ajam.20221002.13
DO  - 10.11648/j.ajam.20221002.13
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 43
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

James Edward Joseph

Retired, Department of Mathematics, Howard University, Washington, USA
Bhamini M. P. Nayar

Department of Mathematics, Morgan State University, Baltimore, USA

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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

Copy | Download

@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}
}

Copy | Download

TY  - JOUR
T1  - Connectedness Generalizations Using the Concept of Adherence Dominators
AU  - James Edward Joseph
AU  - Bhamini M. P. Nayar
Y1  - 2022/04/23
PY  - 2022
N1  - https://doi.org/10.11648/j.ajam.20221002.13
DO  - 10.11648/j.ajam.20221002.13
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 43
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  -

Copy | Download