Mildly generalized closed sets, almost normal and mildly normal spaces

Abstract

The aim of this paper is to introduce and study the class of mildly generalized closed sets, which is properly placed between the classes of strongly generalized closed sets due to Sundaram and Pushpalatha in 2001 and weakly generalized closed sets due to Sundaram and Nagaveni in 1998. The relations with other notions directly or indirectly connected with generalized closed are investigated. Moreover we use it to obtain new characterizations and preservation theorems of almost normal spaces due to Singal and Arya and mildly normal spaces due to Singal and Singal, respectively.

Introduction

Continuity, compactness, connectedness and separation axioms on topological spaces, as important and basic subjects in studies of General Topology and several branches of mathematics, have been researched by many mathematicians. Recent progress in the study of characterizations and generalizations of these subjects has been done by means of several “generalized closed sets”. The first step of generalizing closed sets was done by Levine in 1970 [16]. He defined a subset A of a topological space (X,τ) to be g-closed if its closure belongs to every open superset of A. He also introduced a class of topological spaces called T_1/2 spaces, i.e. topological space (X,τ) is T_1/2 if every g-closed set is closed, and then Dunham and Levine, in [7], [8], further studied some properties of T_1/2 spaces. The notion of g-closed sets has been studied extensively in recent years by many topologist because g-closed sets are only natural generalization of closed sets. More importantly, they also suggested several new properties of topological spaces. The study of generalized closed sets has produced some new separations which are between T₀ and T₁ such as T_1/2 [16], semi-pre-T_1/2 [5], T_p [28] and T_3/4 [6]. Some of these have been found to be useful in computer science and digital topology (see [12], [13], [14], [15], for example). So the study of g-closed sets will give the possible applications in computer graphics [13], [14], [15] and quantum physics [9]. In a recent work El Naschie derived quantum gravity from set theory [29].

As the weak forms of g-closed sets, the notions of rg-closed and weakly g-closed sets were introduced and studied by Palaniappan and Rao [21] and Sundaram and Nagaveni [27], respectively. The notion of rg-closed sets was further studied by Arockiarani and Balachadran [2] and Park et al. [22]. More recently, Sundaram and Pushpalatha [28] introduced the notion of strongly g-closed sets, which is implied by that of closed sets and implies that of g-closed sets. The purpose of this paper is to introduce a new class of generalized closed sets, namely mildly g-closed sets, which is properly placed between the classes of strongly g-closed and weakly g-closed sets. The relations with other notions directly or indirectly connected with g-closed are investigated. Moreover as applications, using the notions of mildly g-closed and mildly g-open sets, we obtain some characterizations of almost normal spaces [24] and mildly normal spaces [26], respectively, and improve the preservation theorems of almost normal and mildly normal spaces established by Singal and Arya [24] and Singal and Singal [26], respectively.