Dense Sets on Bigeneralized Topological Spaces

Int. Journal of Math. Analysis, Vol. 7, 2013, no. 21, 999 - 1003 HIKARI Ltd, www.m-hikari.com

Supunnee Sompong

Department of Mathematics and Statistics Faculty of Science and Technology Sakon Nakhon Rajabhat University Sakon Nakhon 47000, Thailand s [email protected]

Copyright c 2013 Supunnee Sompong. 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.

Abstract

The purpose of this paper is to introduce the concept and some fundamental properties of dense sets on bigeneralized topological spaces.

Keywords: bigeneralized topological spaces, dense sets

1 Introduction

In 2010, the notion of bigeneralized topological spaces and the concepts of weakly functions were introduced by C. Boonpok [1]. He also investigated some of their characterizations. In 2012, S. Sompong and B. Rodjanadid [2], introduced a deﬁnition of dense sets in biminimal structure spaces and study some fundamental of their properties. In 2013, S. Sompong and S. Muangchan [3, 4], introduced the deﬁnition of boundary and exterior set on bigeneralized topological spaces and the concepts of basic properties. In this paper, we introduce the concept of dense sets on bigeneralized topological spaces and study some properties.

2 Preliminaries

Let X be a nonempty set and g be a collection of subsets of X. Then g is called a generalized topology (brieﬂy GT)onX if ∅∈g and if Gi ∈ g for i ∈ I = ∅ then G = i∈I Gi ∈ g. 1000 S. Sompong

By (X, g), we denote a nonempty set X with a generalized topology g on X and it is called a generalized topological space (briefly GTS) on X. The elements of G are called g-open sets and the complements are called g-closed sets. Proposition 2.1. [5] Let (X, g) be a generalized topological space. For subsets A and B of X, the following properties holds; 1. Cl(X \ A)=X \ Int(A) and Int(X \ A)=X \ Cl(A); 2. If X \ A ∈ g, then Cl(A)=A and if A ∈ g, then Int(A)=A; 3. If A ⊆ B, then Cl(A) ⊆ Cl(B) and Int(A) ⊆ Int(B); 4. A ⊆ Cl(A) and Int(A) ⊆ A; 5. Cl(Cl(A)) = Cl(A) and Int(Int(A)) = Int(A). 1 2 Definition 2.2. [1] Let X be a nonempty set and gX , gX be generalized 1 2 topologies on X. A triple (X, gX ,gX ) is called a bigeneralized topological space (briefly BGTS). 1 2 Definition 2.3. [1] A subset A of a bigeneralized topological space (X, gX ,gX) i j i j is called gX gX − closed if A = g Cl(g Cl(A)), where i, j=1 or 2 and i = j. The i j i j complement of gX gX − closed set is called gX gX − open. 1 2 Proposition 2.4. [1] Let (X, gX ,gX) be a bigeneralized topological space. i j i j If A and B are gX gX − closed, then A ∩ B is gX gX − closed. 1 2 Proposition 2.5. [1] Let (X, gX ,gX ) be a bigeneralized topological space. Then i j 1 2 i j A is gX gX − open subset of (X, gX ,gX) if and only if A = g Int(g Int(A)). 1 2 Proposition 2.6. [1] Let (X, gX ,gX) be a bigeneralized topological space. i j i j If A and B are gX gX − open, then A ∪ B is gX gX − open. 1 2 Definition 2.7. [4] Let (X, gX ,gX ) be a bigeneralized topological space, A be i j a subset of X and x ∈ X. We called x is gX gX -exterior point of A if i j i j x ∈ g Int(g Int(X \ A)). We denote the set of all gX gX -exterior point of A by gExtij(A) where i, j =1, 2 and i = j. i j From definition we have gExtij(A)=X \ g Cl(g Cl(A)). 1 2 Definition 2.8. [3] Let (X, gX ,gX ) be a bigeneralized topological space, A be a subset of X and x ∈ X. We called x is (i, j) − gX − boundary point of A if x ∈ giCl(gjCl(A)) ∩ giCl(gjCl(X\A)). We denote the set of all (i, j) − gX −boundary point of A by gBdrij(A) where i, j =1, 2 and i = j. i j i j From definition we have gBdrij(A)=g Cl(g Cl(A)) ∩ g Cl(g Cl(X\A)). 1 2 Theorem 2.9. [3] Let (X, gX ,gX ) be a bigeneralized topological space and A be a subset of X. Then for any i, j =1, 2 and i = j, we have; i j 1. A is gX gX − closed if and only if gBdrij(A) ⊆ A. i j 2. A is gX gX − open if and only if gBdrij(A) ⊆ (X\A). Dense sets on bigeneralized topological spaces 1001

3 Dense Sets on Bigeneralized Topological Spaces

In this section, we introduce the concept of dense sets on bigeneralized topological spaces and study some fundamental of their properties.

1 2 Deﬁnition 3.1. Let (X, gX ,gX) be a bigeneralized topological spaces, A be i j i j a subset of X. A is called gX gX -dense set in X if X = g Cl(g Cl(A)), where i, j =1, 2 and i = j.

1 2 Example 3.2. Let X = {1, 2, 3}. Deﬁne g-structures gX and gX on X as fol- 1 2 lows: gX = {∅, {1}, {2}, {1, 2}, {1, 3}, {2, 3},X} and gX = {∅, {1}, {3}, {1, 3}, {2, 3},X}. Then g1Cl(g2Cl({1, 3})) = X and g2Cl(g1Cl({1, 3})) = X. g1Cl(g2Cl({2, 3})) = {2, 3} and g2Cl(g1Cl({2, 3})) = {2, 3}. 1 2 2 1 Hence {1, 3} is gX gX -dense and gX gX -dense set in X. i j But {2, 3} is not gX gX -dense set in X, where i, j =1, 2 and i = j.

1 2 Theorem 3.3. Let (X, gX ,gX ) be a bigeneralized topological space and A be i j a subset of X. A is gX gX -dense set in X if and only if gExtij(A)=∅, where i, j =1, 2 and i = j.

Proof. Let A be a subset of X. i j (=⇒) Suppose that A is gX gX -dense set in X. Then we have i j gExtij(A)=X \ g Cl(g Cl(A)) = ∅, where i, j =1, 2 and i = j.

i j (⇐=) Assume that gExtij(A)=∅. Thus X \ g Cl(g Cl(A)) = ∅, i j i j it follows that g Cl(g Cl(A)) = X. Therefore A is gX gX -dense set in X, where i, j =1, 2 and i = j.

1 2 Theorem 3.4. Let (X, gX ,gX ) be a bigeneralized topological spaces and A be a i j i j subset of X. If A is gX gX -dense set in X then for any non-empty gX gX -closed subset F of X, where i, j =1, 2 and i = j such that A ⊆ F, we have F = X.

i j i j Proof. Suppose that A is gX gX -dense set in X and F is gX gX -closed subset i j of X, where i, j =1, 2 and i = j such that A ⊆ F. Since A is gX gX -dense set i j i j in X, X = g Cl(g Cl(A)). By assumption, F is gX gX -closed and A ⊆ F, it follows that, X = giCl(gjCl(A)) ⊆ giCl(gjCl(F )) = F. Hence F = X.

i j i j Note: By Theorem 3.4 if A is gX gX -dense set in X. Then only X is gX gX - closed set in X such that containing A.

i j Remark 3.5. The Theorem 3.4 is not true if F is not gX gX -closed. We can be seen from the following example. 1002 S. Sompong

1 2 Example 3.6. Let X = {1, 2, 3}. Deﬁne g-structures gX and gX on X as fol- 1 2 lows: gX = {∅, {1, 3}, {2, 3},X} and gX = {∅, {1}, {2}, {1, 2}, {1, 3}, {2, 3},X}. 1 2 1 2 Thus g Cl(g Cl({1, 2})) = X, and we have {1, 2} is gX gX -dense set in X. But 1 2 {1, 2} is not gX gX -closed set in X. 1 2 Theorem 3.7. Let (X, gX ,gX ) be a bigeneralized topological spaces and A be i j a subset of X. If for any non-empty gX gX -closed subset F of X such that i j A ⊆ F, then F = X if and only if G ∩ A = ∅ for any non-empty gX gX -open subset G of X, where i, j =1, 2 and i = j. 1 2 Proof. Let (X, gX ,gX ) be a bigeneralized topological spaces and A ⊆ X. i j (=⇒) Assume that if for any non-empty gX gX -closed subset F of X such that i j A ⊆ F, then F = X. Suppose that G∩A = ∅ for some a non-empty gX gX -open i j subset G of X, where i, j =1, 2 and i = j. Thus A ⊆ X \ G. Since G is gX gX - i j open, X \ G is gX gX -closed. By assumption, we have X \ G = X. Therefore i j G = ∅, this is contradiction. Hence G ∩ A = ∅ for any non-empty gX gX -open subset G of X, where i, j =1, 2 and i = j.

i j (⇐=) Assume that G ∩ A = ∅ for any non-empty gX gX -open subset G of X i j and F is a non-empty gX gX -closed subset of X, where i, j =1, 2 and i = j i j such that A ⊆ F. Suppose that F = X. Thus X \ F is a non-empty gX gX -open subset of X. By assumption, we have (X \ F ) ∩ A = ∅. This is contradiction with A ⊆ F. Therefore F = X.

1 2 Corollary 3.8. Let (X, gX ,gX ) be a bigeneralized topological spaces and A be i j a subset of X. If A is gX gX -dense set in X. Then G∩A = ∅ for any non-empty i j gX gX -open subset G of X, where i, j =1, 2 and i = j. i j Proof. Let A be gX gX -dense set in X. By Theorem 3.4 and 3.7, we have i j G ∩ A = ∅ for any non-empty gX gX -open subset G of X, where i, j =1, 2 and i = j. 1 2 Theorem 3.9. Let (X, gX ,gX ) be a bigeneralized topological spaces and A be i j i j a subset of X. Then gBdrij(A)=g Cl(g Cl(X\A)) if and only if A is gX gX - dense set in X, where i, j =1, 2 and i = j. 1 2 Proof. Let (X, gX ,gX ) be a bigeneralized topological spaces and A ⊆ X. i j (=⇒) Assume that gBdrij(A)=g Cl(g Cl(X\A)). Thus giCl(gjCl(A)) ∩ giCl(gjCl(X\A)) = giCl(gjCl(X\A)). It follows that, X\(giInt(gjInt(A)) = giCl(gjCl(X\A)) ⊆ giCl(gjCl(A)). Then we have, X\(giCl(gjCl(A)) ⊆ giInt(gjInt(A) ⊆ giCl(gjCl(A)). i j Hence A is gX gX -dense set in X, where i, j =1, 2 and i = j.

i j (⇐=) Assume that A is gX gX -dense set in X. Thus we have, i j i j i j gBdrij(A)=g Cl(g Cl(A)) ∩ g Cl(g Cl(X\A)) = g Cl(g Cl(X\A)). Dense sets on bigeneralized topological spaces 1003

1 2 Theorem 3.10. Let (X, gX ,gX ) be a bigeneralized topological spaces and A be i j i j a subset of X. Then A is gX gX -open and gX gX -dense set in X if and only if gBdrij(A)=X\A, where i, j =1, 2 and i = j.

1 2 Proof. Let (X, gX ,gX ) be a bigeneralized topological spaces and A ⊆ X. i j i j (=⇒) Assume that A is gX gX -open and gX gX -dense set in X. Then by Theorem 2.9 and Theorem 3.9, we have giCl(gjCl(X\A)) ⊆ (X\A). i j Hence gBdrij(A)=g Cl(g Cl(X\A)) = (X\A), where i, j =1, 2 and i = j.

(⇐=) Assume that gBdrij(A)=X\A. i j Then by Theorem 2.9 we have, A is gX gX -open set in X. i j i j And also gBdrij(A)=X\A = X\(g Int(g Int(A)) = g Cl(g Cl(X\A)). i j Thus by Theorem 3.9 we have, A is gX gX -dense set in X, where i, j =1, 2 and i = j.

1 2 Example 3.11. Let X = {1, 2, 3}. Deﬁne g-structures gX and gX on X as fol- 1 2 lows: gX = {∅, {1, 3}, {2, 3},X} and gX = {∅, {1}, {2}, {1, 2}, {1, 3}, {2, 3},X}. Thus g1Cl(g2Cl({1, 3})) = X, and g1Int(g2Int({1, 3})) = {1, 3}. 1 2 1 2 Hence we have {1, 3} is gX gX -open and gX gX -dense set in X. Then We also have gBdr12({1, 3})={2}.

Acknowledgement. The author would like to thank Sakon Nakhon Ra- jabhat University and National Research Council of Thailand (NRCT) for ﬁnancial support.

References

[1] C. Boonpok, Weakly Open Functions on Bigeneralized Topological Spaces, Int. Journal of Math. Analysis, 4 (18) (2010), 891–897.

[2] S. Sompong and B. Rodjanadid, Dense Set in Biminimal Structure Spaces, Int. Journal of Math. Analysis, 6 (6) (2012), 279–283.

[3] S. Sompong and S. Muangchan, Boundary Set on Bigeneralized Topolog- ical Spaces, Int. Journal of Math. Analysis, 7 (2) (2013), 85–89.

[4] S. Sompong and S. Muangchan, Exterior Set on Bigeneralized Topological Spaces, Int. Journal of Math. Analysis, 7 (15) (2013), 719–723.

[5] W.K. Min, Almost Continuity on Generalized Topological Spaces , Acta Math. Hungar., 125 (2009), 121–125.

Received: January, 2013