Some New Sets and Topologies in Ideal Topological Spaces

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