P S -Regular Sets in Topology and Generalized Topology

Abstract

We define and study a new class of regular sets called P S -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 P S -regular sets. Semiconnectedness is characterized by using P S -regular sets. P S -continuity and almost P S -continuity are introduced and investigated.

Ankit Gupta 1 and Ratna Dev Sarma 2

Academic Editor:Peter R. Massopust

1, Department of Mathematics, University of Delhi, Delhi 110007, India
2, Department of Mathematics, Rajdhani College, University of Delhi, Delhi 110015, India

Received 23 May 2014; Revised 25 August 2014; Accepted 26 August 2014; 14 September 2014

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [1-3].

Due to this, investigations of these sets have gained momentum in the recent days. Csaszar has already provided an umbrella study for generalized open sets in his latest papers [4-7]. In this paper, we introduce and study a new class of sets, called [figure omitted; refer to PDF] -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 [figure omitted; refer to PDF] -regular sets with other existing classes such as semiopen sets, regular open sets, [figure omitted; refer to PDF] -sets, [figure omitted; refer to PDF] -sets, [figure omitted; refer to PDF] -sets, and [figure omitted; refer to PDF] -sets has been studied. A characterization of semiconnectedness is also provided using [figure omitted; refer to PDF] -regular sets. Moreover, [figure omitted; refer to PDF] -regular sets, where [figure omitted; refer to PDF] , of a generalized topological space are studied using [figure omitted; refer to PDF] -regular sets. In the last two sections, [figure omitted; refer to PDF] -regularity is studied in the domain of general topological spaces. Here several decompositions of regular open sets and regular closed sets are provided using [figure omitted; refer to PDF] -regular sets. In the last section, [figure omitted; refer to PDF] -continuity and almost [figure omitted; refer to PDF] -continuity are defined and interrelationship of almost [figure omitted; refer to PDF] -continuity with other existing mappings such as [figure omitted; refer to PDF] -map, graph mapping, almost precontinuity, and almost [figure omitted; refer to PDF] -continuity is investigated.

2. Preliminaries

First we recall some definitions and results to be used in the paper.

Definition 1 (see [6]).

Let [figure omitted; refer to PDF] be a nonempty set. A collection [figure omitted; refer to PDF] of subsets of [figure omitted; refer to PDF] is called a generalized topology (in brief, [figure omitted; refer to PDF] ) on [figure omitted; refer to PDF] if it is closed under arbitrary unions. The ordered pair [figure omitted; refer to PDF] is called generalized topological space (in brief, [figure omitted; refer to PDF] ).

Since an empty union amounts to the empty set, [figure omitted; refer to PDF] always belongs to [figure omitted; refer to PDF] . However, [figure omitted; refer to PDF] need not be a member of [figure omitted; refer to PDF] . The members of [figure omitted; refer to PDF] are called [figure omitted; refer to PDF] -open while the complements of [figure omitted; refer to PDF] -open sets are called [figure omitted; refer to PDF] -closed. The largest [figure omitted; refer to PDF] -open set contained in a set [figure omitted; refer to PDF] is called the interior of [figure omitted; refer to PDF] and is denoted by [figure omitted; refer to PDF] , whereas the smallest [figure omitted; refer to PDF] -closed set containing [figure omitted; refer to PDF] is called the closure of [figure omitted; refer to PDF] and is denoted by [figure omitted; refer to PDF] . For [figure omitted; refer to PDF] , we have, [figure omitted; refer to PDF] , [figure omitted; refer to PDF] and whenever [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] . These properties will be used in the text without any further mention.

Remark 2.

Although Csaszar provided the definition of a generalized topology, similar notions existed prior to Csaszar's work also. Peleg [8] in 1984 defined a similar structure which he named "semitopology" on [figure omitted; refer to PDF] . The corresponding "semitopological closure" is found to be monotone, enlarging, and idempotent. A subfamily [figure omitted; refer to PDF] of the power set of [figure omitted; refer to PDF] which is closed under nonempty intersection has been studied in the literature under the name of "intersection structure" [9]; the corresponding topped intersection structure is known as closure system.

Definition 3.

Let [figure omitted; refer to PDF] be a [figure omitted; refer to PDF] . Then a subset [figure omitted; refer to PDF] of [figure omitted; refer to PDF] is called

(i) [figure omitted; refer to PDF] semiopen [7] if [figure omitted; refer to PDF] ;

(ii) [figure omitted; refer to PDF] - [figure omitted; refer to PDF] -open [7] if [figure omitted; refer to PDF] ;

(iii): [figure omitted; refer to PDF] -preopen [7] if [figure omitted; refer to PDF] ;

(iv) [figure omitted; refer to PDF] - [figure omitted; refer to PDF] -open [7] if [figure omitted; refer to PDF] ;

(v) [figure omitted; refer to PDF] -set [10] if [figure omitted; refer to PDF] ;

(vi) regular open (resp ., regular closed ) [11] if [figure omitted; refer to PDF] (resp., [figure omitted; refer to PDF] ).

The complement of a [figure omitted; refer to PDF] -semiopen (resp., [figure omitted; refer to PDF] - [figure omitted; refer to PDF] -open, [figure omitted; refer to PDF] -preopen, and [figure omitted; refer to PDF] - [figure omitted; refer to PDF] -open) set is called [figure omitted; refer to PDF] -semiclosed (resp., [figure omitted; refer to PDF] - [figure omitted; refer to PDF] -closed, [figure omitted; refer to PDF] -preclosed, and [figure omitted; refer to PDF] - [figure omitted; refer to PDF] -closed).

The families of regular open, regular closed, and [figure omitted; refer to PDF] -sets on [figure omitted; refer to PDF] are denoted by [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , respectively.

The intersection of all [figure omitted; refer to PDF] -semiclosed sets containing a set [figure omitted; refer to PDF] is called the semiclosure of [figure omitted; refer to PDF] and is denoted by [figure omitted; refer to PDF] . Dually, the semi-interior of [figure omitted; refer to PDF] is defined to be the union of all [figure omitted; refer to PDF] -semiopen sets contained in [figure omitted; refer to PDF] and is denoted by [figure omitted; refer to PDF] .

Theorem 4 (see [5]).

In a [figure omitted; refer to PDF] with [figure omitted; refer to PDF] , one has

(i) [figure omitted; refer to PDF] ,

(ii) [figure omitted; refer to PDF] ,

where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] denote the semi-interior and semiclosure of [figure omitted; refer to PDF] , respectively.

3. [figure omitted; refer to PDF] -Regular Sets in Generalized Topological Spaces

Although the following discussion is provided for generalized topological space, it is also valid for topological spaces as every topological space is a generalized topological space as well.

Definition 5.

A subset [figure omitted; refer to PDF] of a generalized topological space [figure omitted; refer to PDF] is said to be [figure omitted; refer to PDF] -regular if [figure omitted; refer to PDF] .

The class of all [figure omitted; refer to PDF] -regular sets in [figure omitted; refer to PDF] is denoted by [figure omitted; refer to PDF] .

Lemma 6.

For a generalized topological space [figure omitted; refer to PDF] , [figure omitted; refer to PDF] and [figure omitted; refer to PDF] both are [figure omitted; refer to PDF] -regular sets.

Proof.

Let [figure omitted; refer to PDF] be a [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] . Now, consider [figure omitted; refer to PDF] . If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] and we are done. If not, then let [figure omitted; refer to PDF] , the largest [figure omitted; refer to PDF] -open set in [figure omitted; refer to PDF] . If [figure omitted; refer to PDF] then again [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -regular. Let, if possible, [figure omitted; refer to PDF] , and then [figure omitted; refer to PDF] is a [figure omitted; refer to PDF] -open set in [figure omitted; refer to PDF] and hence [figure omitted; refer to PDF] is again a [figure omitted; refer to PDF] -open set containing [figure omitted; refer to PDF] , which leads to a contradiction. Hence [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -regular. Similarly, it can be shown that [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -regular.

Thus one can say that the family of [figure omitted; refer to PDF] -regular sets [figure omitted; refer to PDF] forms an [figure omitted; refer to PDF] -structure [12]. However, the family [figure omitted; refer to PDF] is not closed under finite union as well as finite intersection.

We have the following example.

Example 7.

Let [figure omitted; refer to PDF] be the set of all real numbers with the usual topology [figure omitted; refer to PDF] . Take [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ; both are [figure omitted; refer to PDF] -regular sets. But [figure omitted; refer to PDF] is not a [figure omitted; refer to PDF] -regular set because [figure omitted; refer to PDF] .

Similarly, take [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ; both are [figure omitted; refer to PDF] -regular sets. But [figure omitted; refer to PDF] is not a [figure omitted; refer to PDF] -regular set because [figure omitted; refer to PDF] .

Theorem 8.

Every [figure omitted; refer to PDF] -regular set is [figure omitted; refer to PDF] -semiopen.

Proof.

Let [figure omitted; refer to PDF] be a [figure omitted; refer to PDF] -regular set. Then [figure omitted; refer to PDF] . Therefore [figure omitted; refer to PDF] . Hence [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -semiopen.

But converse of this result is not true in general. We have the following example.

Example 9.

Let [figure omitted; refer to PDF] with [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] is a [figure omitted; refer to PDF] -semiopen set because [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -open. But [figure omitted; refer to PDF] is not [figure omitted; refer to PDF] -regular because [figure omitted; refer to PDF] .

Remark 10.

The notions of [figure omitted; refer to PDF] -regular sets and [figure omitted; refer to PDF] -open sets are independent of each other. Similarly, [figure omitted; refer to PDF] -regular sets are different from [figure omitted; refer to PDF] - [figure omitted; refer to PDF] -open, [figure omitted; refer to PDF] -preopen, and [figure omitted; refer to PDF] - [figure omitted; refer to PDF] -open sets as well.

In Example 9 mentioned above, it is clear that [figure omitted; refer to PDF] is a [figure omitted; refer to PDF] -open set; thus it is [figure omitted; refer to PDF] - [figure omitted; refer to PDF] -open, [figure omitted; refer to PDF] -preopen, and [figure omitted; refer to PDF] - [figure omitted; refer to PDF] -open. But [figure omitted; refer to PDF] is not [figure omitted; refer to PDF] -regular.

Example 11.

Let [figure omitted; refer to PDF] be the set of all real numbers with its usual topology [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] is a [figure omitted; refer to PDF] -preopen set in [figure omitted; refer to PDF] as [figure omitted; refer to PDF] . But [figure omitted; refer to PDF] is not [figure omitted; refer to PDF] -regular because [figure omitted; refer to PDF] .

Thus [figure omitted; refer to PDF] -regular sets are independent of [figure omitted; refer to PDF] -preopen sets.

Example 12.

Let [figure omitted; refer to PDF] with [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] is a [figure omitted; refer to PDF] -regular set. But [figure omitted; refer to PDF] is neither [figure omitted; refer to PDF] -open nor [figure omitted; refer to PDF] -preopen or [figure omitted; refer to PDF] - [figure omitted; refer to PDF] -open in [figure omitted; refer to PDF] because [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

Hence from the above examples, it is clear that [figure omitted; refer to PDF] -regular sets are independent of [figure omitted; refer to PDF] -open sets, [figure omitted; refer to PDF] -preopen sets, and [figure omitted; refer to PDF] - [figure omitted; refer to PDF] -open sets as well.

Theorem 13.

Let [figure omitted; refer to PDF] be a generalized topological space. Then a subset [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -regular if it is both [figure omitted; refer to PDF] -semiopen and [figure omitted; refer to PDF] -semiclosed.

Proof.

Let [figure omitted; refer to PDF] be [figure omitted; refer to PDF] -semiopen as well as [figure omitted; refer to PDF] -semiclosed. Then [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Therefore [figure omitted; refer to PDF] . Hence [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -regular.

Theorem 14.

Every regular open set is [figure omitted; refer to PDF] -regular.

Proof.

Let [figure omitted; refer to PDF] be a regular open set. Therefore [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -open and hence [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -semiopen. Again as [figure omitted; refer to PDF] is regular open, we have [figure omitted; refer to PDF] . Thus [figure omitted; refer to PDF] . Hence [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -semiclosed. Therefore by Theorem 13, it follows that [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -regular.

But the converse of the above result is not true in general. We have the following example.

Example 15.

In Example 12, [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -regular. But [figure omitted; refer to PDF] is not regular open as [figure omitted; refer to PDF] .

Theorem 16.

Every regular closed set is [figure omitted; refer to PDF] -regular.

Proof.

Let [figure omitted; refer to PDF] be regular closed. Therefore [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -closed and [figure omitted; refer to PDF] -semiclosed. As [figure omitted; refer to PDF] is regular closed, [figure omitted; refer to PDF] and hence [figure omitted; refer to PDF] . Thus [figure omitted; refer to PDF] is both [figure omitted; refer to PDF] -semiopen and [figure omitted; refer to PDF] -semiclosed. Hence [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -regular in [figure omitted; refer to PDF] .

But the converse of the above theorem is not true in general. We have the following example.

Example 17.

Let [figure omitted; refer to PDF] be the set of all real numbers with the usual topology [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] is not a regular closed set. But it is [figure omitted; refer to PDF] -regular.

In this section, we provide two interesting characterizations of [figure omitted; refer to PDF] -regular sets.

Theorem 18.

Let [figure omitted; refer to PDF] be a generalized topological space. Then a subset [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -regular if and only if [figure omitted; refer to PDF] .

Proof.

For a generalized topological space, we know that [figure omitted; refer to PDF] Therefore, we have [figure omitted; refer to PDF] Consider [figure omitted; refer to PDF] . But [figure omitted; refer to PDF] ; thus [figure omitted; refer to PDF] . Therefore [figure omitted; refer to PDF] . Thus, [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] , as [figure omitted; refer to PDF] (shown above). Thus [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -regular if and only if [figure omitted; refer to PDF] .

Corollary 19.

A set [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -regular if and only if it is both [figure omitted; refer to PDF] -semiopen and [figure omitted; refer to PDF] -semiclosed.

Proof.

From the above result, it follows that if [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -regular set then [figure omitted; refer to PDF] . Hence [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -semiclosed. Thus the result follows in the light of Theorems 8 and 13.

Corollary 20.

If a set [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -regular then its complement is also [figure omitted; refer to PDF] -regular.

The above implications may be represented diagrammatically in Figure 1.

Figure 1: [figure omitted; refer to PDF]

We complete this section by providing few interesting decompositions and results using [figure omitted; refer to PDF] -regular sets.

Definition 21 (see [13]).

Let [figure omitted; refer to PDF] be a generalized topological space and [figure omitted; refer to PDF] . [figure omitted; refer to PDF] is said to be [figure omitted; refer to PDF] -regular (resp., [figure omitted; refer to PDF] -regular , [figure omitted; refer to PDF] -regular , and [figure omitted; refer to PDF] -regular ) if [figure omitted; refer to PDF] (resp., [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] ).

The following theorems provide the relationships between [figure omitted; refer to PDF] -regular sets where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] -regular sets. Here it may be mentioned that [figure omitted; refer to PDF] -regular sets and [figure omitted; refer to PDF] -regular sets are nothing but regular open sets and regular closed sets defined in Definition 3.

Theorem 22.

Let [figure omitted; refer to PDF] be a GTS and [figure omitted; refer to PDF] . Then every [figure omitted; refer to PDF] -regular set, where [figure omitted; refer to PDF] , is [figure omitted; refer to PDF] -regular.

Proof.

Let [figure omitted; refer to PDF] be [figure omitted; refer to PDF] -regular. Then [figure omitted; refer to PDF] . Hence by Theorem 14, it is [figure omitted; refer to PDF] -regular. Similarly if [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -regular, then [figure omitted; refer to PDF] . Hence by Theorem 16, [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -regular as well. Further, we know that a set is [figure omitted; refer to PDF] -regular if and only if it is an [figure omitted; refer to PDF] -regular [13] and a set is [figure omitted; refer to PDF] -regular if and only if it is [figure omitted; refer to PDF] -regular [13]. Therefore every [figure omitted; refer to PDF] -regular set, where [figure omitted; refer to PDF] , is [figure omitted; refer to PDF] -regular.

Theorem 23.

Let [figure omitted; refer to PDF] be a GTS and [figure omitted; refer to PDF] . Then the following hold.

(i) [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -regular if and only if [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -regular and [figure omitted; refer to PDF] -closed if and only if [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -regular and [figure omitted; refer to PDF] -closed.

(ii) [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -regular if and only if [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -regular and [figure omitted; refer to PDF] -closed if and only if [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -regular and [figure omitted; refer to PDF] -closed.

Proof.

(i) If [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -regular then it is clear that [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -regular and [figure omitted; refer to PDF] -closed which implies that [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -closed as well. Suppose [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -regular and [figure omitted; refer to PDF] -closed. Then [figure omitted; refer to PDF] ; that is, [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Hence [figure omitted; refer to PDF] which implies that [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -regular.

(ii) It is similar to (i).

Definition 24.

A subset [figure omitted; refer to PDF] of a generalized topological space [figure omitted; refer to PDF] is called [figure omitted; refer to PDF] -closed [14] if [figure omitted; refer to PDF] whenever [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -open in [figure omitted; refer to PDF] , and the finite unions of regular open sets are said to be [figure omitted; refer to PDF] -open.

Theorem 25.

Let [figure omitted; refer to PDF] be a GTS and [figure omitted; refer to PDF] . Then the following hold.

(i) Every [figure omitted; refer to PDF] -regular set is [figure omitted; refer to PDF] -closed set.

(ii) [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -regular if and only if [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -open and [figure omitted; refer to PDF] -regular.

Proof.

(i) Let [figure omitted; refer to PDF] be a [figure omitted; refer to PDF] -regular set. Let [figure omitted; refer to PDF] be any [figure omitted; refer to PDF] -open set in [figure omitted; refer to PDF] containing [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Since [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -regular and every [figure omitted; refer to PDF] -regular set is [figure omitted; refer to PDF] -semiclosed in [figure omitted; refer to PDF] , therefore, [figure omitted; refer to PDF] . Hence [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -closed.

(ii) If [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -regular then clearly [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -open and [figure omitted; refer to PDF] -regular set. Conversely, let [figure omitted; refer to PDF] be [figure omitted; refer to PDF] -open and [figure omitted; refer to PDF] -regular set. Hence [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -semiclosed; that is, [figure omitted; refer to PDF] ; that is, [figure omitted; refer to PDF] . Since [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -open set, therefore [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -preopen and hence [figure omitted; refer to PDF] . Thus [figure omitted; refer to PDF] . Hence [figure omitted; refer to PDF] is a [figure omitted; refer to PDF] -regular set.

An application of [figure omitted; refer to PDF] -regular set is the characterization of semiconnectedness [15].

Definition 26.

A generalized topological space [figure omitted; refer to PDF] is said to be semidisconnected if there exist two [figure omitted; refer to PDF] -semiopen sets [figure omitted; refer to PDF] and [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ; otherwise it is called semiconnected .

Theorem 27.

A generalized topological space is semiconnected if and only if it does not contain any proper [figure omitted; refer to PDF] -regular set.

Proof.

Suppose [figure omitted; refer to PDF] is a proper [figure omitted; refer to PDF] -regular set in [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] is also a [figure omitted; refer to PDF] -regular set. Therefore [figure omitted; refer to PDF] and [figure omitted; refer to PDF] both are [figure omitted; refer to PDF] -semiopen sets such that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Hence [figure omitted; refer to PDF] is semidisconnected.

Conversely, let [figure omitted; refer to PDF] be a semidisconnected space. Then there exist two [figure omitted; refer to PDF] -semiopen sets [figure omitted; refer to PDF] and [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] , a [figure omitted; refer to PDF] -semiclosed set as well. Hence [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -semiopen and [figure omitted; refer to PDF] -semiclosed both. Therefore [figure omitted; refer to PDF] is a proper [figure omitted; refer to PDF] -regular set in [figure omitted; refer to PDF] .

Our investigations on [figure omitted; refer to PDF] -regular sets so far have been in the domain of generalized topological spaces. In the remaining part of our paper, we study interrelationships of the notion of [figure omitted; refer to PDF] -regularity with other existing topological notions. Hence, from now onwards, we confine our investigation to the domain of topological spaces only. Also, following the usual convention of topology, we denote interior and closure of a set [figure omitted; refer to PDF] by [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , respectively, in our discussion. Since a topological space is also a generalized topological space, therefore all the results of this section so far are also valid for topological spaces.

4. Some Decompositions Using [figure omitted; refer to PDF] -Regular Sets

Apart from semiopen and semiclosed sets, there are several other important generalized forms of open sets and closed sets in topology such as [figure omitted; refer to PDF] -set, [figure omitted; refer to PDF] -set, [figure omitted; refer to PDF] -set, [figure omitted; refer to PDF] -set, and [figure omitted; refer to PDF] set. In this section, we study [figure omitted; refer to PDF] -regular sets in the light of these sets. We also provide some interesting decompositions of regular open and regular closed sets using the notion of [figure omitted; refer to PDF] -regular sets.

First of all, we provide the following definitions for topological spaces.

Definition 28.

Let ( [figure omitted; refer to PDF] ) be a topological space. A subset [figure omitted; refer to PDF] is said to be

(i) a [figure omitted; refer to PDF] -set [16] if [figure omitted; refer to PDF] ;

(ii) a [figure omitted; refer to PDF] -set [16] if there is an open set [figure omitted; refer to PDF] and a [figure omitted; refer to PDF] -set [figure omitted; refer to PDF] in [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] ;

(iii): a [figure omitted; refer to PDF] -set [10] if there exists [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , such that [figure omitted; refer to PDF] ;

(iv) a [figure omitted; refer to PDF] -set [17] if [figure omitted; refer to PDF] .

Theorem 29.

Every [figure omitted; refer to PDF] -regular set is [figure omitted; refer to PDF] -set and [figure omitted; refer to PDF] -set.

Proof.

Let [figure omitted; refer to PDF] be a [figure omitted; refer to PDF] -regular set. Then [figure omitted; refer to PDF] . Therefore [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] because [figure omitted; refer to PDF] .

Thus [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -set. Also [figure omitted; refer to PDF] is an [figure omitted; refer to PDF] -set as well because [figure omitted; refer to PDF] .

But the converse is not always true. We have the following examples.

Example 30.

Let [figure omitted; refer to PDF] with topology [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] is a [figure omitted; refer to PDF] -set because [figure omitted; refer to PDF] . But [figure omitted; refer to PDF] is not [figure omitted; refer to PDF] -regular because [figure omitted; refer to PDF] is not semiopen as [figure omitted; refer to PDF] .

Example 31.

Let [figure omitted; refer to PDF] with topology [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] is an [figure omitted; refer to PDF] -set as [figure omitted; refer to PDF] . But [figure omitted; refer to PDF] is not a [figure omitted; refer to PDF] -regular set because [figure omitted; refer to PDF] .

Corollary 32.

Every [figure omitted; refer to PDF] -regular set is a [figure omitted; refer to PDF] -set, [figure omitted; refer to PDF] -set, and [figure omitted; refer to PDF] -set.

Proof.

It is because every [figure omitted; refer to PDF] -regular set is [figure omitted; refer to PDF] -set and [figure omitted; refer to PDF] -set.

Now, we provide some decompositions of regular closed sets using [figure omitted; refer to PDF] -regular sets.

Theorem 33.

For a topological space, the following are equivalent:

(i) [figure omitted; refer to PDF] is regular closed;

(ii) [figure omitted; refer to PDF] is closed and [figure omitted; refer to PDF] -regular;

(iii): [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -closed and [figure omitted; refer to PDF] -regular;

(iv) [figure omitted; refer to PDF] is preclosed and [figure omitted; refer to PDF] -regular.

Proof.

[figure omitted; refer to PDF] ii [figure omitted; refer to PDF] : they are proved in Theorem 16.

[figure omitted; refer to PDF] ii [figure omitted; refer to PDF] iii [figure omitted; refer to PDF] : let [figure omitted; refer to PDF] be a closed set; then it is [figure omitted; refer to PDF] -closed because every closed set is [figure omitted; refer to PDF] -closed.

[figure omitted; refer to PDF] iii [figure omitted; refer to PDF] iv [figure omitted; refer to PDF] : they are obvious because every [figure omitted; refer to PDF] -closed set is preclosed as well.

[figure omitted; refer to PDF] iv [figure omitted; refer to PDF] : let [figure omitted; refer to PDF] be a preclosed and [figure omitted; refer to PDF] -regular set. Then [figure omitted; refer to PDF] is semiopen. Thus [figure omitted; refer to PDF] . Since [figure omitted; refer to PDF] is preclosed, therefore [figure omitted; refer to PDF] as well. Hence [figure omitted; refer to PDF] and is hence regular closed.

Now, we proceed to provide decompositions of regular open sets using [figure omitted; refer to PDF] -regular sets.

Theorem 34.

For a topological space, the following are equivalent:

(i) [figure omitted; refer to PDF] is regular open;

(ii) [figure omitted; refer to PDF] is open and [figure omitted; refer to PDF] -regular;

(iii): [figure omitted; refer to PDF] is preopen and [figure omitted; refer to PDF] -regular;

(iv) [figure omitted; refer to PDF] is preopen and semiclosed.

Proof.

[figure omitted; refer to PDF] ii [figure omitted; refer to PDF] : they are proved earlier in Theorem 14.

[figure omitted; refer to PDF] ii [figure omitted; refer to PDF] iii [figure omitted; refer to PDF] : let [figure omitted; refer to PDF] be an open set. Then it is preopen because every open set is preopen.

[figure omitted; refer to PDF] iii [figure omitted; refer to PDF] iv [figure omitted; refer to PDF] : as every [figure omitted; refer to PDF] -regular set is semiclosed therefore [figure omitted; refer to PDF] is semiclosed.

[figure omitted; refer to PDF] iv [figure omitted; refer to PDF] : let [figure omitted; refer to PDF] be a preopen and semiclosed set. Thus [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ; that is, [figure omitted; refer to PDF] . Hence [figure omitted; refer to PDF] . Thus [figure omitted; refer to PDF] is regular open.

Theorem 35.

A set [figure omitted; refer to PDF] is regular open if and only if it is [figure omitted; refer to PDF] -open and [figure omitted; refer to PDF] -regular.

Proof.

We have already proved in Theorem 34 that every regular open set is [figure omitted; refer to PDF] -regular and open and hence [figure omitted; refer to PDF] -open. Conversely, let [figure omitted; refer to PDF] be [figure omitted; refer to PDF] -regular; then it is an [figure omitted; refer to PDF] -set by Theorem 29. Hence [figure omitted; refer to PDF] is regular open because a set is regular open if and only if it is [figure omitted; refer to PDF] -open and [figure omitted; refer to PDF] -set [10].

5. [figure omitted; refer to PDF] -Continuity and Almost [figure omitted; refer to PDF] -Continuity

We first recall the following definitions.

Definition 36.

A function [figure omitted; refer to PDF] is said to be [figure omitted; refer to PDF] -map [18] (resp., almost continuous [19], almost [figure omitted; refer to PDF] -continuous [20], almost semicontinuous [21], and almost precontinuous [18]) if [figure omitted; refer to PDF] is regular open, (resp., open, [figure omitted; refer to PDF] -set, semiopen, and preopen) for every regular open set [figure omitted; refer to PDF] in [figure omitted; refer to PDF] .

Now we define [figure omitted; refer to PDF] -continuity and almost [figure omitted; refer to PDF] -continuity in the following way.

Definition 37.

(a) A mapping [figure omitted; refer to PDF] is said to be [figure omitted; refer to PDF] -continuous at a point [figure omitted; refer to PDF] if for every neighbourhood [figure omitted; refer to PDF] of [figure omitted; refer to PDF] there exists a [figure omitted; refer to PDF] -regular neighbourhood [figure omitted; refer to PDF] of [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] .

(b) A mapping [figure omitted; refer to PDF] is said to be almost [figure omitted; refer to PDF] -continuous at a point [figure omitted; refer to PDF] if for every neighbourhood [figure omitted; refer to PDF] of [figure omitted; refer to PDF] there exists a [figure omitted; refer to PDF] -regular neighbourhood [figure omitted; refer to PDF] of [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] .

A mapping [figure omitted; refer to PDF] is said to be [figure omitted; refer to PDF] -continuous (resp., almost [figure omitted; refer to PDF] -continuous ) if it is [figure omitted; refer to PDF] -continuous (resp., almost [figure omitted; refer to PDF] -continuous) at each point [figure omitted; refer to PDF] of [figure omitted; refer to PDF] .

Since every regular open set is open, therefore every [figure omitted; refer to PDF] -continuous mapping is almost [figure omitted; refer to PDF] -continuous.

Theorem 38.

For a mapping [figure omitted; refer to PDF] , the following are equivalent:

(i) [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -continuous;

(ii) inverse image of every open subset of [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -regular;

(iii): inverse image of every closed subset of [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -regular.

Proof.

[figure omitted; refer to PDF] ii [figure omitted; refer to PDF] : let [figure omitted; refer to PDF] be any open subset of [figure omitted; refer to PDF] and let [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] . Therefore there exists a [figure omitted; refer to PDF] -regular subset [figure omitted; refer to PDF] in [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Thus [figure omitted; refer to PDF] ; therefore [figure omitted; refer to PDF] is a [figure omitted; refer to PDF] -regular neighbourhood of [figure omitted; refer to PDF] . Hence [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -regular.

[figure omitted; refer to PDF] ii [figure omitted; refer to PDF] iii [figure omitted; refer to PDF] : let [figure omitted; refer to PDF] be any closed subset of [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] is open and therefore [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -regular; that is, [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -regular. Hence [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -regular.

[figure omitted; refer to PDF] iii [figure omitted; refer to PDF] : let [figure omitted; refer to PDF] be open neighbourhood of [figure omitted; refer to PDF] ; therefore [figure omitted; refer to PDF] is closed, and consequently [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -regular. Thus [figure omitted; refer to PDF] is also [figure omitted; refer to PDF] -regular and hence [figure omitted; refer to PDF] (say). Then [figure omitted; refer to PDF] is a [figure omitted; refer to PDF] -regular neighbourhood of [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] .

Theorem 39.

For a mapping [figure omitted; refer to PDF] , the following are equivalent:

(i) [figure omitted; refer to PDF] is almost [figure omitted; refer to PDF] -continuous at [figure omitted; refer to PDF] ;

(ii) for every regular open neighbourhood [figure omitted; refer to PDF] of [figure omitted; refer to PDF] , there is a [figure omitted; refer to PDF] -regular neighbourhood [figure omitted; refer to PDF] of [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] .

Proof.

[figure omitted; refer to PDF] ii [figure omitted; refer to PDF] : if [figure omitted; refer to PDF] is almost [figure omitted; refer to PDF] -continuous at [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is a regular open neighbourhood of [figure omitted; refer to PDF] , then there is a [figure omitted; refer to PDF] -regular neighbourhood [figure omitted; refer to PDF] of [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] .

[figure omitted; refer to PDF] ii [figure omitted; refer to PDF] : it is obvious.

Theorem 40.

For a mapping [figure omitted; refer to PDF] , the following are equivalent:

(i) [figure omitted; refer to PDF] is almost [figure omitted; refer to PDF] -continuous;

(ii) inverse image of every regular open subset of [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -regular;

(iii): inverse image of every regular closed subset of [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -regular;

(iv) for each point [figure omitted; refer to PDF] of [figure omitted; refer to PDF] and for each regular open neighbourhood [figure omitted; refer to PDF] of [figure omitted; refer to PDF] , there is a [figure omitted; refer to PDF] -regular neighbourhood [figure omitted; refer to PDF] of [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] .

Proof.

The proof is the same as Theorem 38.

Theorem 41.

If [figure omitted; refer to PDF] is a [figure omitted; refer to PDF] -continuous (resp., almost [figure omitted; refer to PDF] -continuous) map then it is semicontinuous (resp., almost semicontinuous).

Proof.

Let [figure omitted; refer to PDF] be any open (resp., regular open) subset in [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] is a [figure omitted; refer to PDF] -regular and hence a semiopen set because [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -continuous (resp., almost [figure omitted; refer to PDF] -continuous). Hence [figure omitted; refer to PDF] is semicontinuous (resp., almost semicontinuous).

Theorem 42.

If [figure omitted; refer to PDF] is an almost [figure omitted; refer to PDF] -continuous map then the following hold:

(i) [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -map if and only if it is almost precontinuous;

(ii) [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -map if and only if it is almost [figure omitted; refer to PDF] -continuous.

Proof.

(i) Let [figure omitted; refer to PDF] be a [figure omitted; refer to PDF] -map and let [figure omitted; refer to PDF] be any regular open subset in [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] is regular open and hence preopen. Thus [figure omitted; refer to PDF] is almost precontinuous.

Conversely, let [figure omitted; refer to PDF] be almost precontinuous and almost [figure omitted; refer to PDF] -regular continuous map. Let [figure omitted; refer to PDF] be any regular open subset of [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] is preopen as well as [figure omitted; refer to PDF] -regular. Hence [figure omitted; refer to PDF] is regular open. Therefore [figure omitted; refer to PDF] is a [figure omitted; refer to PDF] -map.

(ii) It is the same as [figure omitted; refer to PDF] i [figure omitted; refer to PDF] .

Remark 43.

From the above theorem we can conclude that a map is almost continuous (a.c.s) if it is almost precontinuous and almost [figure omitted; refer to PDF] -continuous.

Theorem 44.

Let [figure omitted; refer to PDF] be a function and let [figure omitted; refer to PDF] be the graph function defined by [figure omitted; refer to PDF] , for every [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] is almost [figure omitted; refer to PDF] -continuous if [figure omitted; refer to PDF] is almost [figure omitted; refer to PDF] -continuous.

Proof.

Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] containing [figure omitted; refer to PDF] . Then, we have [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Since [figure omitted; refer to PDF] is almost [figure omitted; refer to PDF] -continuous, there exists a [figure omitted; refer to PDF] -regular set [figure omitted; refer to PDF] of [figure omitted; refer to PDF] containing [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] . Therefore we obtain [figure omitted; refer to PDF] and hence [figure omitted; refer to PDF] is almost [figure omitted; refer to PDF] -continuous.

Acknowledgments

The authors express sincere thanks to the referees of the paper. The suggestions provided by one of the referees have greatly improved the presentation of the paper. This work is a part of a research work financed by the University Grant Commission (India).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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