%0 Journal Article
%T -Regular Sets in Topology and Generalized Topology
%A Ankit Gupta
%A Ratna Dev Sarma
%J Journal of Mathematics
%D 2014
%I Hindawi Publishing Corporation
%R 10.1155/2014/274592
%X We define and study a new class of regular sets called -regular sets. Properties of these sets are investigated for topological spaces and generalized topological spaces. Decompositions of regular open sets and regular closed sets are provided using -regular sets. Semiconnectedness is characterized by using -regular sets. -continuity and almost -continuity are introduced and investigated. 1. Introduction In general topology, repeated applications of interior and closure operators give rise to several different new classes of sets. Some of them are generalized form of open sets while few others are the so-called regular sets. These classes are found to have applications not only in mathematics but even in diverse fields outside the realm of mathematics [1¨C3]. Due to this, investigations of these sets have gained momentum in the recent days. Cs¨¢sz¨¢r has already provided an umbrella study for generalized open sets in his latest papers [4¨C7]. In this paper, we introduce and study a new class of sets, called -regular sets, using semi-interior and semiclosure operators. Initially, we define them for a broader class, that is, for generalized topological spaces and discuss their various properties. Interrelationship of -regular sets with other existing classes such as semiopen sets, regular open sets, -sets, -sets, -sets, and -sets has been studied. A characterization of semiconnectedness is also provided using -regular sets. Moreover, -regular sets, where , of a generalized topological space are studied using -regular sets. In the last two sections, -regularity is studied in the domain of general topological spaces. Here several decompositions of regular open sets and regular closed sets are provided using -regular sets. In the last section, -continuity and almost -continuity are defined and interrelationship of almost -continuity with other existing mappings such as -map, graph mapping, almost precontinuity, and almost -continuity is investigated. 2. Preliminaries First we recall some definitions and results to be used in the paper. Definition 1 (see [6]). Let be a nonempty set. A collection of subsets of is called a generalized topology (in brief, ) on if it is closed under arbitrary unions. The ordered pair is called generalized topological space (in brief, ). Since an empty union amounts to the empty set, always belongs to . However, need not be a member of . The members of are called -open while the complements of -open sets are called -closed. The largest -open set contained in a set is called the interior of and is denoted by , whereas the smallest
%U http://www.hindawi.com/journals/jmath/2014/274592/