Idealization of Some Topological Concepts

An approach is followed here to generate a new topology on a set from an ideal and a family of subsets of . The so-obtained topology is related to other known topologies on . The cases treated here include the one when is taken, then the case when is considered. The approach is open to apply to other choices of . As application, some known results are obtained as corollaries to those results appearing here. In the last part of this work, some ideal-continuity concepts are studied, which originate from some previously known terms and results. 1. Introduction The interest in the idealized version of many general topological properties has grown drastically in the past 20 years. In this work, no particular paper will be referred to except where it is needed and encountered. However, many symbols, definitions, and concepts used here are as in [1]. 2. Open Sets via a Family and an Ideal Recall that an ideal on a set is a family of subsets of , that is, (the power set of ), such that is closed under finite union, and if and , then (heredity property). An ideal topological space is a triple , where is a set, is a topology on , and is an ideal on . Let be an ideal topological space. The family and forms a base for a topo y on finer than [1]. As a start, this concept will be put in a more general setting as follows. Definition 2.1. Let be an ideal topological space, and let be a family of subsets of .(a)A set is called an -open set if for each , there exist and such that and , or equivalently . The family of all -open subsets is written .(b)The topology on generated by the subbase is denoted by . Remark 2.2. (1) The family needs not form, in general, a topology on . (2) In the case where , it is clear that , and for the case . Proposition 2.3. Let be an ideal topological space and let . Let and . Let be the topology generated by the subbase , then . Proof. Note that , and therefore . Now let , then for each , there exists with and such that . This means that where for each . Thus, since . Proposition 2.4. Let be an ideal topological space. If denotes the topology on generated by the subbase , then . Proof. To show that , first note that and therefore and . This implies that . Next, consider the base and for . Let , say for some and some . If , then , and so there exist such that , where , that is, , and hence . This shows that . If is a topological space, we let (resp., ) denote the interior of (resp., the closure of ) in . A subset of is called semiopen if , and is called an set if . The family of all -sets forms a topology on finer than . Example 2.5. Let be