The primary goal is to characterize Locally H-closed spaces (LHC), by conditions on the remainders of their extensions. These spaces are also characterized using subspaces and their extensions as well. Characterizing these classes of spaces using the remainders of the subspaces in their extensions provide characterizations of them in terms of their boundaries. Recently, the authors have proved that these results give necessary and sufficient conditions for the space to be compact A number of equivalences are proved for Hausdorff (Urysohn) [regular] spaces. These results lead to similar characterizations of Locally Urysohn-closed (LUC) as well as Locally regular-closed (LRC) spaces. Some of these equivalent properties generalize a number of existing results on these topics. In the present article it is shown that if X is a Hausdorff LHC space then each closed set is an intersection of regularly open sets as well as each closed set is an intersection of semi-closed neighborhoods. In 1969 Porter and Thomas had shown that in a Hausdorff space a locally H-closed subspace is the intersection of an open set and a closed set. In this article, it is shown that a space X is LHC if and only if every nonempty proper regularly closed subset of X is LHC.
| Published in | American Journal of Applied Mathematics (Volume 10, Issue 2) |
| DOI | 10.11648/j.ajam.20221002.14 |
| Page(s) | 51-58 |
| 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), 2024. Published by Science Publishing Group |
H-closed Extensions, Locally H-closed, θ-closure, u-closure, s-closure, θ-rigid
| [1] | R. F. Dickman Jr. and J. R. Porter, θ-closed subsets of Hausdorff spaces, Pacific J. Math. Vol. 59 (1975), 407- 415. |
| [2] | R. F. Dickman Jr. and J. R. Porter, θ-perfect and θ- absolutely closed functions, Illinois J. Math. Vol. 21 (1977), 42-60. |
| [3] | M. S. Espelie and J. E. Joseph, Some properties of θ- closure, Canad. J. Math., Vol. 33, No. 1 (1981), 142-149. |
| [4] | M. S. Espelie, J. E. Joseph and M. H. Kwack, Applications of the u-closure operator, Proc. Amer. Math. Soc. Vol. 83 No. 1 (1981), 167-174. |
| [5] | S. Fomin, Extensions of Topological Spaces, Ann. Math. Vol. 44 (1943), 471-480. |
| [6] | M. Ganster, On strongly s-regular spaces, Glas. Mat. Ser. III. Vol. 25 No. 1 (1990), 195-201. |
| [7] | M. Girou, Properties of Locally H-Closed Spaces, Proc. Amer. Math. Soc. Vol. 113 N0. 1 (1991), 287-295. |
| [8] | L L. Herrington, Characterizations of Regular-closed Spaces, Math. Chronicle 5 (1977), 168-178. |
| [9] | S. Illiads and S. Fomin, The method of centered systems in the theory of topological spaces, Uspelchi Mat. Nauk 21 (1966), 47-76 = Russian Math. Surveys 21 (1966), 37-62. |
| [10] | J. E. Joseph, On Urysohn-closed and minimal Urysohn spaces, Proc, Amer. Math. Soc., 68 (1978), 235-242. |
| [11] | J. E. Joseph, Multifunctions and cluster sets, Proc. Amer. Math. Soc. Vol. 74 (1979), 329-337. |
| [12] | J. E. Joseph, Some remarks on θ-rigidity, Kyungpook Math. Journal Vol. 20 No. 2 (1980), 245-250. |
| [13] | 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). |
| [14] |
M. Kat tov, On H-closed extensions of topological spaces,  asopis Pst. Mat. Fys. Vol. 72 (1947), 17-32.
|
| [15] | B. M. P. Nayar, Minimal sequential Hausdorff spaces, International J. Math. and Math. Sci. Vol. No. 22 April (2004), 1169-1177. |
| [16] | B. M. P. Nayar, Some classes of generalized completely regular spaces, International J. of Pure and Applied Mathematics Vol. 44 No. 4 (2008), 609-626. |
| [17] | F. Obreanu, Espaces localement absolument fermés, Ann. Acad. Pepub. Pop. Române Sect. Sti. Fiz. Chim. Ser. A Vol. 3 (1950), 375-394. |
| [18] | J. R. Porter, On locally H-closed spaces, Proc.Londn Math. Soc. Vol. 3 (20) (1970), 193 -204. |
| [19] | J. R. Porter and J. Thomas, On H-closed and minimal Hausdorff spaces, Trans. Amer. Math Soc. Vol. 138 (1969), 159-170. |
| [20] |
J. R. Porter and M. L. Tikoo, On Kat tov spaces, Canad. Math. Bull. Vol. 32 (1989), 425-433.
|
| [21] | J. R. Porter and R. G. Woods, Extensions and Absolutes of Hausdorff Spaces, Springer Verlag, New York 1988. |
| [22] | M. L. Tikoo, Remainders of H-Closed Extensions, Topology Appl. Vol 10 (1986), 117-128. |
| [23] |
N. V. Veli ko, H-Closed Topological Spaces, Mat. Sb. Vol. 70 (112) (1966), 98-112: Amer. Math. Soc. Transl. Vol. (2) 78 (1968), 103-118.
|
APA Style
James Edward Joseph, Bhamini M. P. Nayar. (2022). Locally H-closed Spaces, Subspaces and Their Extensions. American Journal of Applied Mathematics, 10(2), 51-58. https://doi.org/10.11648/j.ajam.20221002.14
ACS Style
James Edward Joseph; Bhamini M. P. Nayar. Locally H-closed Spaces, Subspaces and Their Extensions. Am. J. Appl. Math. 2022, 10(2), 51-58. doi: 10.11648/j.ajam.20221002.14
AMA Style
James Edward Joseph, Bhamini M. P. Nayar. Locally H-closed Spaces, Subspaces and Their Extensions. Am J Appl Math. 2022;10(2):51-58. doi: 10.11648/j.ajam.20221002.14
Share This Article
Twitter
Linked In
Facebook
Copy
Download@article{10.11648/j.ajam.20221002.14,
author = {James Edward Joseph and Bhamini M. P. Nayar},
title = {Locally H-closed Spaces, Subspaces and Their Extensions},
journal = {American Journal of Applied Mathematics},
volume = {10},
number = {2},
pages = {51-58},
doi = {10.11648/j.ajam.20221002.14},
url = {https://doi.org/10.11648/j.ajam.20221002.14},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20221002.14},
abstract = {The primary goal is to characterize Locally H-closed spaces (LHC), by conditions on the remainders of their extensions. These spaces are also characterized using subspaces and their extensions as well. Characterizing these classes of spaces using the remainders of the subspaces in their extensions provide characterizations of them in terms of their boundaries. Recently, the authors have proved that these results give necessary and sufficient conditions for the space to be compact A number of equivalences are proved for Hausdorff (Urysohn) [regular] spaces. These results lead to similar characterizations of Locally Urysohn-closed (LUC) as well as Locally regular-closed (LRC) spaces. Some of these equivalent properties generalize a number of existing results on these topics. In the present article it is shown that if X is a Hausdorff LHC space then each closed set is an intersection of regularly open sets as well as each closed set is an intersection of semi-closed neighborhoods. In 1969 Porter and Thomas had shown that in a Hausdorff space a locally H-closed subspace is the intersection of an open set and a closed set. In this article, it is shown that a space X is LHC if and only if every nonempty proper regularly closed subset of X is LHC.},
year = {2022}
}
TY - JOUR T1 - Locally H-closed Spaces, Subspaces and Their Extensions AU - James Edward Joseph AU - Bhamini M. P. Nayar Y1 - 2022/04/23 PY - 2022 N1 - https://doi.org/10.11648/j.ajam.20221002.14 DO - 10.11648/j.ajam.20221002.14 T2 - American Journal of Applied Mathematics JF - American Journal of Applied Mathematics JO - American Journal of Applied Mathematics SP - 51 EP - 58 PB - Science Publishing Group SN - 2330-006X UR - https://doi.org/10.11648/j.ajam.20221002.14 AB - The primary goal is to characterize Locally H-closed spaces (LHC), by conditions on the remainders of their extensions. These spaces are also characterized using subspaces and their extensions as well. Characterizing these classes of spaces using the remainders of the subspaces in their extensions provide characterizations of them in terms of their boundaries. Recently, the authors have proved that these results give necessary and sufficient conditions for the space to be compact A number of equivalences are proved for Hausdorff (Urysohn) [regular] spaces. These results lead to similar characterizations of Locally Urysohn-closed (LUC) as well as Locally regular-closed (LRC) spaces. Some of these equivalent properties generalize a number of existing results on these topics. In the present article it is shown that if X is a Hausdorff LHC space then each closed set is an intersection of regularly open sets as well as each closed set is an intersection of semi-closed neighborhoods. In 1969 Porter and Thomas had shown that in a Hausdorff space a locally H-closed subspace is the intersection of an open set and a closed set. In this article, it is shown that a space X is LHC if and only if every nonempty proper regularly closed subset of X is LHC. VL - 10 IS - 2 ER -
Information
Email: