全部 标题 作者
关键词 摘要

OALib Journal期刊
ISSN: 2333-9721
费用:99美元

查看量下载量

相关文章

更多...

Some New Sets and Topologies in Ideal Topological Spaces

DOI: 10.1155/2013/973608

Full-Text   Cite this paper   Add to My Lib

Abstract:

An ideal topological space is a triplet ( , , ), where is a nonempty set, is a topology on , and is an ideal of subsets of . In this paper, we introduce -perfect, -perfect, and -perfect sets in ideal spaces and study their properties. We obtained a characterization for compatible ideals via -perfect sets. Also, we obtain a generalized topology via ideals which is finer than using -perfect sets on a finite set. 1. Introduction and Preliminaries The contributions of Hamlett and Jankovic [1–4] in ideal topological spaces initiated the generalization of some important properties in general topology via topological ideals. The properties like decomposition of continuity, separation axioms, connectedness, compactness, and resolvability [5–9] have been generalized using the concept of ideals in topological spaces. By a space , we mean a topological space with a topology defined on on which no separation axioms are assumed unless otherwise explicitly stated. For a given point in a space , the system of open neighborhoods of is denoted by . For a given subset of a space and are used to denote the closure of and interior of , respectively, with respect to the topology. A nonempty collection of subsets of a set is said to be an ideal on , if it satisfies the following two conditions: (i) If and , then ; (ii) If and , then . An ideal topological space (or ideal space) means a topological space with an ideal defined on . Let be a topological space with an ideal defined on . Then for any subset of for every is called the local function of with respect to and . If there is no ambiguity, we will write or simply for . Also, defines a Kuratowski closure operator for the topology (or simply ) which is finer than . An ideal on a space is said to be codense ideal if and only if . is always a proper subset of . Also, if and only if the ideal is codense. Lemma 1 ([see 12]). Let be a space with and being ideals on , and let and be two subsets on . Then(i) ; (ii) ; (iii) ( is a closed subset of );(iv) ; (v) ; (vi) ;(vii)for every . Definition 2 (see [3]). Let be a space with an ideal on . One says that the topology is compatible with the ideal , denoted by , if the following holds, for every : if for every , there exists a such that , then . Definition 3. A subset of an ideal space is said to be(i) -closed [3] if ,(ii)*-dense-in-itself [10] if ,(iii) -open [11] if ,(iv)almost -open [12] if ,(v) -dense [7] if ,(vi)almost strong - -open [13] if ,(vii)*-perfect [10] if ,(viii)regular -closed [14] if ,(ix)an -set [15] if . Theorem 4 ([3]). Let be a space with an ideal on . Then the

References

[1]  T. R. Hamlett and D. Jankovic, “Ideals in general topology,” General Topology and Applications, pp. 115–125, 1988.
[2]  T. R. Hamlett and D. Jankovic, “Ideals in topological spaces and the set operatory,” Bollettino dell'Unione Matematica Italiana, vol. 7, pp. 863–874, 1990.
[3]  D. Jankovic and T. R. Hamlett, “New topologies from old via ideals,” The American Mathematical Monthly, vol. 97, pp. 295–310, 1990.
[4]  D. Jankovic and T. R. Hamlett, “Compatible extensions of ideals,” Bollettino della Unione Matematica Italiana, vol. 7, no. 6, pp. 453–465, 1992.
[5]  G. Aslim, A. Caksu Guler, and T. Noiri, “On decompositions of continuity and some weaker forms of continuity via idealization,” Acta Mathematica Hungarica, vol. 109, no. 3, pp. 183–190, 2005.
[6]  F. G. Arenas, J. Dontchev, and M. L. Puertas, “Idealization of some weak separation axioms,” Acta Mathematica Hungarica, vol. 89, no. 1-2, pp. 47–53, 2000.
[7]  J. Dontchev, M. Ganster, and D. Rose, “Ideal resolvability,” Topology and Its Applications, vol. 93, no. 1, pp. 1–16, 1999.
[8]  E. Ekici and T. Noiri, “Connectedness in ideal topological spaces,” Novi Sad Journal of Mathematics, vol. 38, no. 2, pp. 65–70, 2008.
[9]  R. L. Newcomb, Topologies which are compact modulo an ideal [Ph.D. dissertation], University of California at Santa Barbara, 1967.
[10]  E. Hayashi, “Topologies defined by local properties,” Mathematische Annalen, vol. 156, no. 3, pp. 205–215, 1964.
[11]  M. E. Abd El-Monsef, E. F. Lashien, and A. A. Nasef, “On -open sets and -continuous functions,” Kyungpook Mathematical Journal, vol. 32, no. 1, pp. 21–30, 1992.
[12]  M. E. Abd El-Monsef, R. A. Mahmoud, and A. A. Nasef, “Almost- -openness and almost- - continuity,” Journal of the Egyptian Mathematical Society, vol. 7, no. 2, pp. 191–200, 1999.
[13]  E. Hatir, A. Keskin, and T. Noiri, “On new decomposition of continuity via idealization,” JP Journal of Geometry & Topology, vol. 1, no. 3, pp. 53–64, 2003.
[14]  A. Keskin, T. Noiri, and ?. Yüksel, “Idealization of a decomposition theorem,” Acta Mathematica Hungarica, vol. 102, no. 4, pp. 269–277, 2004.
[15]  A. Keskin, T. Noiri, and S. Yuksel, “fI-sets and decomposition of RIC-continuity,” Acta Mathematica Hungarica, vol. 104, no. 4, pp. 307–313, 2004.

Full-Text

Contact Us

service@oalib.com

QQ:3279437679

WhatsApp +8615387084133