Regular mildly generalized closed and regular mildly generalized open sets in bitopological spaces

Introduction

A triple(X, τ 1 , τ 2 ) where X is a non-empty set and τ 1 , τ 2 are topologies on X is called bitopological space. J. C. Kelly [5] initiated the systematic study of such spaces in 1963. In 1986, T. Fukutake [3] introduced the concept of generalized closed sets in bitopological spaces and after that several authors turned their attention towards generalizations of various concepts of topology by considering bitopological spaces. M. A. Jalic [4] , N. Nagaveni [8] , I. Arockiarani [1] and M. Sheik John [11] introduced (i, j)-pre-open, (i, j)-wg-closed, (i, j)-rg-closed and (i, j)g * -closed sets in bitopological spaces. Also R. S. Wali and Nirani Laxmi [12, 13] introduce and studied the properties of RMG-closed and RMG-open sets in topological space. The purpose of this paper to introduced and investigate the concept of (τ i , τ j )-RMG-closed and (τ i , τ j )-RMG-open sets which are introduced in a bitopological space by analogy with RMG-closed and RMG-open sets in bitopological space.

Preliminaries

Throughout this paper the space X always means(X, τ 1 , τ 2 ) bitopological spaces on which no separation axioms are assumed unless explicitly stated. Let A be a subset of a space X. by τ iint(A), τ i -cl(A) we shall denote the interior and closure of A⊂X with respect to the topology τ i for i=1, 2. By (i, j) means pair of topologies (τ i , τ j ) and X-A or A c denotes the complement of A in X. Now we recall the following known definitions and results that are used in our work; Definition 2.1: A subset A of a topological space X is called (i) Pre-open [7] , if A⊆int(cl(A)) and pre-closed if cl(int(A))⊆A.

Definition 2.2:

A subset A of a topological space X is called (i) Generalized closed (briefly g-closed) [6] if cl(A) ⊆U whenever A⊆U and U is open in X. (ii) Weakly generalized closed (briefly wg-closed) [8] if cl(int(A)) ⊆U whenever A⊆U and U is open in X.

(iii) Strongly generalized closed (briefly g*-closed) [11] if cl(A) ⊆U whenever A⊆U and U is g-open in X.

(iv) Mildly generalized closed (briefly mildly g-closed) [10] if cl(int(A)) ⊆U whenever A⊆U and U is g-open in X.

(v) Regular weakly closed (briefly rw-closed) [2] if cl(A)⊆U whenever A⊆U and U is regular semi open in X.

(vi) Regular generalized closed (briefly rg-closed) [9] if cl(A)⊆U whenever A⊆U and U is regular open set in X.

(vii) Regular Mildly Generalized closed(briefly RMG-closed) [12] if cl(int(A)) ⊆U whenever A⊆U and U is rg-open in X. The complements of above all closed sets are their respective open sets in the same topological space X. Remark 3.17: (i, j)-g * -closed sets and (i, j)-RMG-closed sets are independent as seen from the following example.

Example 3.18: Let X={a, b, c, d}, τ 1 ={X, ∅, {a}, {b}, {a, b}, {a, b, c}} and τ 2 ={X, ∅, {a}, {b, c}, {a, b, c}}. Then the subset {c} is (1, 2)-RMG-closed set, but not (1, 2)-g * -closed set. Also the subsets {b, d} and (a, b, d} are (1, 2)-g * -closed sets, but not (1, 2)-RMG-closed set in the bitopological space(X, τ 1 , τ 2 ).

Theorem 3.19:

If A be a (i, j)-RMG-closed subset of (X, τ 1 , τ 2 ), then A is (i, j)-wg-closed, but converse is not true.

Proof: Let A be a (i, j)-RMG-closed subset of (X, τ 1 , τ 2 ). Let G∈O(X, τi) be such that A⊆G. Since O(X, τ i )⊆RGO(X, τ i ), we have G∈RGO(X, τi). Then by hypothesis, τ j -cl(τ i -int(A)⊆G. Therefore A is (i, j)-mildly-closed. Remark 3.21: (i, j)-RMG-closed sets and (i, j)-rw-closed sets are independent as seen from the following example.

Example 3.22: Let X={a, b, c, d}, τ 1 ={X, ∅, {a}, {b}, {a, b}, {b, c}, {a, b, c}} and τ 2 ={X, ∅, {a}, {b, c}, {a, b, c}}. Then the subsets {a, b}, {a, c}, {b, d}, {a, b, c} and {a, b, d} are (1, 2)-rw-closed sets, but not (1, 2)-RMG-closed sets. Also the subset {c} is (1, 2)-RMG-closed set, but not (1, 2)-rw-closed set in the bitopological space (X, τ 1 , τ 2 ).

Remark 3.23: (i, j)-RMG-closed sets and (i, j)-rg-closed sets are independent as seen from the following example. Remark 3.28: The union of two (i, j)-RMG-closed sets is generally not a (i, j)-RMG-closed set as seen from the following example.

Example 3.29: Let X={a, b, c, d}, τ 1 ={X, ∅, {a}, {b, c}, {a, b, c}} and τ 2 ={X, ∅, {a}, {b}, {a, b}, {b, c}, {a, b, c}}. Then the subsets {b} and {c} are (1, 2)-RMG-closed sets, but {b}∪{c}={b, c} is not a (1, 2)-RMG-closed set in bitopological space (X, τ 1 , τ 2 ).

Remarks 3.30:

The family DRMG(X, τ 1 , τ 2 ) is generally not equal to the family DRMG(X, τ 2 , τ 1 ) as seen from the following example.

Example 3.31: Let X={a, b, c d},1={X, ∅,{a},{b},{a, b}, {a, b, c}}, 2={X, ∅,{a},{b, c}, {a, b, c}}. Then DRMG(X, τ 1 , τ 2 )= {X, ∅, {c}, {d}, {a, d}, {c, d}, {a, c, d}, {b, c, d}} and DRMG(X,2,1)= {X, ∅, {b}, {c}, {d}, {a, d}, {b, d},{c, d}, {a, b, d} {a, c, d}, {b, c, d}}. Therefore DRMG(X, τ 1 , τ 2 )≠ DRMG(X, τ 2 , τ 1 ).

Theorem 3.32: If τ 1 ⊆ τ 2 and RGO(X, τ 1 ) ⊆ RGO(X, τ 2 ) in(X, τ 1 , τ 2 ), then DRMG(X, τ 1 , τ 2 ) ⊇DRMG(X, τ 2 , τ 1 ).

Proof: Let A be a (τ 2 , τ 1 )-RMG-closed set and G be an τ 1 -rg-open set containing A. By the assumption

Theorem 3.33: Let (i, j)∈{1, 2}. For each point x of (X, 1, 2), a singleton {x} is τ i -rg-closed or {x} c is (τ i , τ j )-RMG-closed.

Proof: Suppose {x} is not τ i -rg-closed. Then {x} c is not i-rg-open. Therefore a i-rg-open set containing {x} c is X only. Also τ jcl(τ i -int({x} c )) ⊆X. Hence {x} c is {i, j}-RMG-closed.

Theorem 3.34: If a subset A of X is ((i, j)-RMG-closed in (X, τ i , τ j )if and only if τ j -cl(τ i -int(A)-A does not contain any nonempty τ i -rg-closed set in (X, τ i , τ j ).

Proof: Suppose that A is a (i, j)-RMG-closed set in (X, τ i , τ j ). We prove the the result by contradiction. Let U be τ i -rg-closed set such that U⊂ τ j -cl(τ i -int(A))-A and U≠ ∅. Then U⊂ τ j -cl(τ i -int(A))∩ A c . Therefore U⊂ τ j -cl(τ i -int(A)) and U⊂ A c is τ i -rgclosed set and A is (i, j)-RMG-closed, τ j -cl(τ i -int(A))⊆ U c . That is U⊆ [τ j -cl(τ i -int(A))] c .Hence U⊆ τ j -cl(τ i -int(A))∩

[τ j -cl(τ i -int(A))] c = ∅ That is U= ∅. This is a contradiction. Hence τ j -cl(τ i -int(A))-A does not contain any non-empty τ i -rgclosed set in (X, τ i , τ j ). Conversely, assume that τ j -cl(τ i -int(A))-A contains no non empty τ i -rg-closed set. Let A⊆U, U is a τ i -rg-closed set. Suppose that τ j -cl(τ i -int(A)) is not contained in U. Then τ j -cl(τ i -int(A))∩ U c is a non empty τ i -rg-closed set and contained in τ j -cl(τ i -int(A))-A, which is contradiction. Therefore τ j -cl(τ i -int(A))⊆U. Hence A is (i, j)-RMG-closed in (X, τ i , τ j ).

Theorem 3.35:

In a bitopological space (X, τ 1 , τ 2 ), RGO(X, τ i ) ⊂{F ⊂X:F c ∈ τ j } if and only if every subset of (X, τ i , τ j ) is a (i, j)-RMG-closed set.

Proof: Suppose that RGO(X, τ i ) ⊂{F ⊂X:F c ∈ τ j }. Let A be any subset of X. G∈ RGO(X, i) be such that A⊂G. Then τ j -cl(τ iint(G))⊂ τ j -cl(A)=G. Also τ j -cl(τ i -int(A)) ⊂ τ j -cl(τ i -int(G))=G. That is τ j -cl(τ i -int(A)) ⊂G. Therefore A is (i, j)-RMG-closed set.

Conversely, suppose that every subset of (X, τ 1 , τ 2 ) is (i, j)-RMG-closed set. Let G∈ RGO(X, τ i ). Since G⊂G and G is(i, j)-RMG-closed, we have τ j -cl(τ i int(G))⊂G.Thus τ j -cl(τ i int(G))⊂ τ j -cl(A) =G and so G is τ j -closed. That is G∈ (F ⊂X:F c ∈ τ j }.

Hence RGO(X, i) ⊂{F ⊂X:F c ∈ τ j }.

Theorem 3.36: If A is a (i, j)-RMG-closed set and A⊂B⊂ τ j -cl(τ i -int(A)), Then B is (i, j)-RMG-closed.

Proof: Let U be a τ j -rg-open set such that B⊂U. As A is (i, j)-RMG-closed set and A⊂G, we have

Therefore B is a (i, j)-RMG closed set.

Theorem 3.37: Let A⊂Y⊂X and suppose that A is (i, j)-RMG-closed in (X,

Proof: τ i-Y be the restriction of τ i to Y. Let G be a τ i-Y -open set such that A⊂G. Since A⊂Y⊂X and Y is τ i -open. By the Lemma 3.26 [12] , G is

Theorem 3.38: In a bitopological space (X, τ 1 , τ 2 ), if RGO(X, τ i )={X, ∅}, Then every subset of (X, τ 1 , τ 2 ) is (i, j)-RMG-closed.

Proof: Let RGO(X,τ i )={X, ∅} in a bitopological space (X, τ 1 , τ 2 ). Let A be any subset of X. To prove that A is an (i, j)-RMGclosed. Suppose A≡ ∅, Then A is (i, j)-RMG-closed. Suppose A≠ ∅, then X is only τ i -rg-open set and τ j -cl(τ i -int(A))⊂X. Hence A is a (i, j)-RMG-closed set.

4.(𝛕 𝐢 , 𝛕 𝐣 )-Rmg-Open Sets And Some Of Their Properties.

In this section, we introduce (i, j)-RMG-open sets in bitopological spaces and study some of their properties.

Definition 4.1: Let i, j∈{1, 2} be fixed integers. In a bitopological space (X, τ 1 , τ 2 ), a subset A⊂X is said to be (τ i , τ j )-Regular Mildly Generalized open (briefly, (i, j)-RMG-open) if A c is (i, j)-RMG-closed. We denote the family of all (i, j)-RMG-open sets in a bitopological space (X, τ 1 , τ 2 ) by D * RMG(τ i , τ j ) or D * RMG(i, j) Proof: Suppose that F is τ i -rg-closed set, F⊆A and F⊆ τ j -int(τ i -cl(A)). Let G be a The converse of the above theorem need not be true as seen from the following example. Theorem 4.16: If a subset A of (X, τ 1 , τ 2 ) is (i, j)-RMG-closed, then τ j -cl(τ i -int(A))-A is (i, j)-RMG-open.

Proof: Let A be a (i, j)-RMG-closed subset in (X, τ 1 , τ 2 ). Let F be a τ i -rg-open set such that F⊆ τ j -cl(τ i -int(A))-A. By Ttheorem 3.34, F= ∅. Therefore F⊆ τ j -int (τ i -cl(τ j -cl(τ i -int(A))-A)) and by Theorem 4.13, τ j -int(τ i -cl(A)) is (i, j)-RMG-open. The converse of the above theorem need not be true as seen from the following example.

Example 4.17: For a subset A={b} in X. X={a, b, c, d}, τ 1 ={X, ∅, {a}, {b}, {a, b}, {a, b, c}} and τ 2 ={X, ∅, {a, b}, {c, d}}. Then τ j -cl(τ i -int(A))-A={a, b}-{b}={a} is (1, 2)-RMG-open but A={b} is not (1, 2)-RMG-closed.

References

1. Arockiarani I. Studies on generalization of generalized closed sets and maps in topological spaces, Ph.D Thesis, Bharathiar University, Coimbatore, 1997.
2. Benchalli SS, Wali RS. On rw-closed sets on topological spaces, Bull. MalaysiaMath. Sci. Soc., 2007; 30:99-110.
3. Fukutake T. On generalize closed sets in bitopological spaces, Bull. Fukuoka Univ. Ed. Part III, 1986; 35:55-60.
4. Jelic MA. decomposition of pairwise continuity, J. Inst. Math. and Comp. Sci., 1990; 3:25-29
5. Kelly JC. Bitopological spaces, Proc. London Math. Soc. 1963; 13:71-81.
6. Levine N. Generalized closed sets in topology, Rend. Circ. Math. Palermo. 1970; 19:89-96.
7. Mashhour AS, Abd ME, El-Monsef, El-Deeb SN. On pre-continuous and mappings and weakly pre-continuous mappings, Proc. Math. Phy. Soc. Egypt. 1982; 53:47-53
8. Nagaveni N. Studies on Generalizations of Homeomorphisms in Topological Spaces, Ph.D.Thesis, Bharathiar University, Coimbatore, 1999.
9. Palaniappan N, Rao KC. Regular generalised closed sets, kyungpook math, J. 1993; 33:211-219.
10. Park JK, Park JH. Mildly generalized closed sets, almost normal and mildly normal spaces, Chaos, Solitions and Fractals, 2004; 20:1103-1111.
11. Sheik John M. A study on generalizations of sets on continuous maps in topological and bitopological spaces, Ph.D. Thesis Bharathiar Univ. Coimbatore, 2002.
12. Wali RS, Nirani Laxmi. On Regular Mildly Generalized (RMG) Closed sets in topological spaces. International Journal of Mathematical Archive. 2016; 7(7):108-114.
13. Wali RS, Nirani Laxmi. On Regular Mildly Generalized (RMG) Open Sets in Topological Spaces, IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Ver. IV 2016; 12(4):93-100.