# Some Separation Axioms Via Ideals

Bollettino dell'Unione Matematica Italiana (2007)

- Volume: 10-B, Issue: 3, page 917-931
- ISSN: 0392-4041

## Access Full Article

top## Abstract

top## How to cite

topSivaraj, D., and Renuka Devi, V.. "Some Separation Axioms Via Ideals." Bollettino dell'Unione Matematica Italiana 10-B.3 (2007): 917-931. <http://eudml.org/doc/290420>.

@article{Sivaraj2007,

abstract = {We introduce a new class of spaces, called Hausdorff modulo $\mathcal\{I\}$ or $T_\{2\}$ mod $\mathcal\{I\}$ spaces with respect to an ideal $\mathcal\{I\}$ which contains the class of all Hausdorff spaces. Characterizations of these spaces are given and their properties are investigated. The concept of compactness modulo an ideal $\mathcal\{I\}$ was introduced by Newcomb in 1967 and studied by Hamlett and Jankovic in 1990. We study the properties of $\mathcal\{I\}$-compact subsets in Hausdorff modulo $\mathcal\{I\}$ spaces and generalize some results of Hamlett and Jankovic. $\mathcal\{I\}$-regular space was introduced by Hamlett and Jankovic in 1994. We further investigate the concept of $\mathcal\{I\}$-regularity with regard to its preservation by functions, subspaces and product.},

author = {Sivaraj, D., Renuka Devi, V.},

journal = {Bollettino dell'Unione Matematica Italiana},

language = {eng},

month = {10},

number = {3},

pages = {917-931},

publisher = {Unione Matematica Italiana},

title = {Some Separation Axioms Via Ideals},

url = {http://eudml.org/doc/290420},

volume = {10-B},

year = {2007},

}

TY - JOUR

AU - Sivaraj, D.

AU - Renuka Devi, V.

TI - Some Separation Axioms Via Ideals

JO - Bollettino dell'Unione Matematica Italiana

DA - 2007/10//

PB - Unione Matematica Italiana

VL - 10-B

IS - 3

SP - 917

EP - 931

AB - We introduce a new class of spaces, called Hausdorff modulo $\mathcal{I}$ or $T_{2}$ mod $\mathcal{I}$ spaces with respect to an ideal $\mathcal{I}$ which contains the class of all Hausdorff spaces. Characterizations of these spaces are given and their properties are investigated. The concept of compactness modulo an ideal $\mathcal{I}$ was introduced by Newcomb in 1967 and studied by Hamlett and Jankovic in 1990. We study the properties of $\mathcal{I}$-compact subsets in Hausdorff modulo $\mathcal{I}$ spaces and generalize some results of Hamlett and Jankovic. $\mathcal{I}$-regular space was introduced by Hamlett and Jankovic in 1994. We further investigate the concept of $\mathcal{I}$-regularity with regard to its preservation by functions, subspaces and product.

LA - eng

UR - http://eudml.org/doc/290420

ER -

## References

top- ABD EL-MONSEF, M.E. - MAHMOUD, R.A. - NASEF, A. A., On quasi $\mathcal{I}$-openness and quasi $\mathcal{I}$-continuity, Tamkang J. Maths., 31 (2000), 101-108. Zbl0986.54024MR1762720
- DONTCHEV, J., On Hausdorff spaces via topological ideals and $\mathcal{I}$-irresolute functions, Annals of the New York Academy of Sciences, 767 (1995), 28-38. Zbl0924.54029MR1462379DOI10.1111/j.1749-6632.1995.tb55891.x
- DONTCHEV, J. - GANSTER, M., On compactness with respect to countable extensions of ideals and the generalized Banach category Theorem, Acta. Math. Hung., 88 (2000), 53-58. Zbl0958.54024MR1780512DOI10.1023/A:1006748410027
- DONTCHEV, J. - GANSTER, M. - ROSE, D., Ideal resolvability, Topology and its Applications, 93 (1999), 1-16. MR1684048DOI10.1016/S0166-8641(97)00257-5
- HAMLETT, T. R. - ROSE, D., *-topological properties, Inter. J. Math. and Math. Sci., 13 (1990), 507-512. MR1068014DOI10.1155/S0161171290000734
- HAMLETT, T. R. - ROSE, D., Local compactness with respect to an ideal, Kyungpook Math. J., 32 (1992), 31-43. Zbl0767.54019MR1170488
- HAMLETT, T.R. - JANKOVIC, D., Compactness with respect to an Ideal, Boll. U.M.I., (7) 4-B (1990), 849-861. Zbl0741.54001MR1086708
- JANKOVIC, D. - HAMLETT, T.R., Compatible Extensions of Ideals, Boll.U.M.I., (7) 6-B (1992), 453-465. Zbl0818.54002MR1191948
- HAMLETT, T. R. - JANKOVIC, D. - ROSE, D., Countable Compactness with respect to an Ideal, Math. Chronicle, 20 (1991), 109-126. Zbl0791.54030MR1137878
- HAMLETT, T. R. - JANKOVIC, D., On weaker forms of paracompactness, countable compactness and Lindelofness, Annals of the New York Academy of Sciences, Papers on General Topology and Applications, 728 (1994), 41-49. Zbl0911.54018MR1467761DOI10.1111/j.1749-6632.1994.tb44132.x
- JANKOVIC, D. - HAMLETT, T.R., New Topologies from old via Ideals, Amer. Math. Monthly, 97, No. 4 (1990), 295-310. Zbl0723.54005MR1048441DOI10.2307/2324512
- KANIEWSKI, J. - PIOTROWSKI, Z., Concerning continuity apart from a meager set, Proc. Amer. Math. Soc., 98 (1986), 324-328. Zbl0606.54008MR854041DOI10.2307/2045706
- KURATOWSKI, K., Topology, Vol I, Academic Press, New York,1966. MR217751
- NASEF, A. A., On Hausdorff spaces via ideals and quasi I-irresolute functions, Chaos, Solitons and Fractals, 14 (2002), 619-625. Zbl1010.54022MR1906922DOI10.1016/S0960-0779(01)00207-7
- NEWCOMB, R. L., Topologies which are compact modulo an ideal, Ph.D. Dissertation, Univ. of Cal. at Santa Barbara, 1967.
- ROSE, D. A. - JANKOVIC, D., On functions having the property of Baire, Real Analysis Exchange, 19 (2) (1993/94), 589-597. Zbl0813.54006MR1282677
- SCARBOROUGH, C. T. - STONE, A. H., Product of Nearly compact Spaces, Trans. Amer. Math. Soc., 124 (1966), 131-147. Zbl0151.30001MR203679DOI10.2307/1994440
- VAIDYANATHASWAMY, R., The Localization Theory in Set Topology, Proc. Indian Acad. Sci., 20 (1945), 51-61. Zbl0061.39308MR10961
- VAIDYANATHASWAMY, R., Set Topology, Chelsea Publishing Company, 1960. MR115151

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.