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SOME SEPARATION PROPERTIES USING a-OPEN SETS

M.K.R.S.VEERA KUMAR

......In this paper some separation properties using a-open sets in topological spaces are defined and their relationships with some other properties are surdied.

1. Introduction

......Throughout this paper by a space X we mean it is a topological space. If A is any subset of a space X, .then cl(A), int(A) and C(A) denote the closure, the interior and the complement of A respectively. ...A susbset A of a space X is called semi-open [6] (resp. a-open [11],. pre-open [10]). if A ` cl(int(A)). (resp. .A ` int(cl(int(A))), .A .` int(cl(A))). .The class of all semi-open (resp. pre-open,. a-open) subsets of a space X is. denoted by. SO(X) (resp. PO(X), .a(X))... The complement of .a. semi-open (resp. pre-open,. a-open) subset .of. a. space. X .is .called. semi-closed .(resp. .pre-closed, a-closed) .set.. ....Scl(A).... (resp. pcl(A), .acl(A)) .denote ..the ..semi-closure. (resp. pre-closure, .a-closure) of the set A... Maheswari and Tapi [9] called a subseet B of a space X as .feebly open. if there is an open set G such that G. ` .B .` .scl(G).. ..Later Jankovic and Reilly [5] observed that feebly open sets are precisely a-open sets.

2.Prereqisites

......Let us recall the following definitions:

......DEFINITION 2.1. A space (X, t) is called

(1) C0 (semi-C0) if, for x,y c X, x g y, there exixts G c t(SO(X)) such that cl(G) (scl(G)) contains only one of x and y but not the other:

(2) C1 (semi-C1) if, for x,y c X, x g y, there exixts G, H c t(SO(X)) such that x c cl(G) (scl(G)), y c cl(H) (scl(H)) but x " cl(H) (scl(H)) and y " cl(G) (scl(G));

(3) w-C0 [3] if 3 ker(x) = f, where ker(x) = 3 {G: x c G c t};................................... ........................x c X

(4) weakly semi-C0 if 3 sker(x) = f, where sker(x) = 3 {G: x c G c SO(X)};.................. ....................................x c X

(5) R0 [2] if cl({x}) ` G whenever x c G c t;

(6) semi-R0 [8] if, for x c G c SO(X), scl({x}) ` G;

(7) weakly R0 [3] if 3 cl({x}) = f;.......................................................................................... .................................x c X

(8) weakly semi-R0 [1] if 3 scl({x}) =f;................................................................................ ...........................................x c X

(9) weakly pre-R0 if 3 pcl({x}) =f;......................................................................................... ..................................x c X

(10) weakly pre-C0 if 3 pker(x) = f, where pker(x) = 3 {G: x c G c PO(X)};.................. ....................................x c X

(11) a-space [4] if every a-open set in it is open.

......Maheswari and Prasad [7] introduced semi-Ti (i=0,1,2) axiom, which is .............. weaker than Ti (i=0,1,2) axiom.

......We use the following sets and classes for counter examples.

......Let X = {a, b, c, d}, Y = {a, b, c}, Z = {a, b, c, d, e, f}.

......Let t1 = {X, f, {b}, {a, b}, {b, c} {a, b, c}}, s1 = {X, f, {a}, {b}, {a, b}},

...........h1 = {Z, f, {a, c, e}, {b, d, f}}. s2 = {Y, f, {a}, {b}, {a, b}, {b, c}},

...........t2 = {X, f, {a}, {b}, {c}, {a, b}, {a, c}, {b, c} {a, b, c}},

...........s = {Y, f, {a}, {b}, {a, b}}.

......THEOREM 2.2 ..1. Every C1 (semi-C1) space is a C0 (semi-C0).

2. Every C0 (C1) space is a semi-C0 (semi-C1).

3. Every R0 space is a weakly R0 [3].

4. Every weakly R0 space is a weakly semi-R0 [1].

5. Every semi-C0 (semi-C1) space is a semi-T0 (semi-T1).

PROOF: Omitted.

......REMARK 2.3. (X, s1) is semi-C0 but not C0. (X, t2) is semi-C1, C0 but not C1.

(Y, s2) is semi-T0 but not semi-C0. (Z, h1) is an a-space but not an a-C0.

3. a-C0, a-C0, Weakly a-C0 and Weakly a-R1 Spaces.

......Now we introduce the following separation properties using a-open sets in spaces.

......DEFINITION 3.1 A space X is called

(1) a-C0 if, for x,y c X, x g y, there exixts G c a(X) such that acl(G) contains only one of x and y but not the other.

(2) a-C1 if, for x,y c X, x g y, there exixts G, H c a(X) such that x c acl(G), y c acl(H) but x " acl(H) and y " acl(G);

(3) weakly a-C0 if 3 aker(x) = f, where aker(x) = 3 {G: x c G c a(X)};.................. ...............................x c X

(4) weakly a-R0 if 3 acl({x}) =f;......................................................................................... ...............................x c X

......THEOREM 3.2 ..1. Every a-C1 space is a-C0.

2. Every a-C0 (a-C1) space is semi-C0(semi-C1).

3. Every weakly a-R0 space is weakly semi-R0 and weakly pre-R0...

4. Every w-C0 space is weakly a-C0.

5. Every weakly a-C0 space is weakly semi-C0 and weakly pre-C0.

6. Every a-C0 (a-C1) space is semi-T0 (semi-T1).

7. Every weakly R0 space is weakly a-R0.

8. Weakly a-R0 ness and weakly a-C0 ness are independent notions.

......REMARK 3.3. .(X, t2) is a-C0 .but .not. a-C1. ..(X, s1) .is. semi-C0, .semi-C1, .but neither a-C1 nor a-C0. (Y, s) is weakly semi-R0 but not weakly a-R0.. (X, t2) is weakly a-C0 but not weakly a-R0. (X, t1) is weakly a-R0 but not weakly a-C0.

......THEOREM 3.4. A space X is weakly a-R0 if and only if aker(x) g X, for each x c X.

PROOF. Necessity - If. there. is.some. x0 c .X .with .aker(x0) = X, ..then X. is the only a-open .set .containing x0. .This .implies. that .x0 c .acl({x}) for every x c. X... Hence

..3 acl({x}) g f, a,contradiction.

x c X

......Sufficiency - If X is not weakly a-R0, then choolse some x0 c.3 aker(X).

.............................................................................................x c X

This implies that every a-neighborhood of x0 contins every point of X. Hence aker(x0) = X.

.....THEOREM 3.5. A space X is weakly a-C0 if and only if for each x c X, there exists a proper a-closed set containing x0.

PROOF. Necessity - Suppose there is some x0 c.X such that X is the only a-closed set containing x0...Let U be any proper a-open subset of X conntaining .a. point. x. ..This implies that .C(U) g X. .Since C(U) is .a-closed, .we have x0 c.C(U). .So x0 c..U...Thus

x0 c.3 ker(x) for any point x of X, a contradiction.

.....x c .X

......Sufficiency - If X is not weakly a-C0, then choose x0 c.3 aker(x). So x0 belongs to

...................................................................................x c .X

aker(x) for .any x c.X. ..This implies that .X .is the only a-open set which contains the point x0, a contradiction.

.....THEOREM 3.6. Every a-C0 (a-C1) space is weakly a-C0.

PROOF. If x, y c X such that x g y, where X is an a-C0 space, then without loss of generality, we can assume that there exists G c a(X) such that x c acl(G) but y " acl(G). This implies that G g f. Hence we can choose some z in G. Now aker(z) 3 aker(y) ` G 3 C(acl(G)) ` (acl(G)) 3 C(acl(G)) = f.

Hence 3 ker(x) = f. Since every a-C1 space is also a-C0, it is also clear that every

........x c .X

a-C1 space is weakly a-C0.

......REMARK 3.7. .The converse of the above theorem need not be true since (Z, h1) is weakly a-C0 but not a-C0. The space (Y, s) is both a-C0 and weakly a-C0 but not a-C1.

.....THEOREM 3.8. The property of being an a-C0 space is not hereditary.

PROOF. Consider the space (Y, s). Let S = {a, c} and s1* be the relative topology on S. It is easy to verify that (Y, s) is a-C0 but its subspace (S, s1*) is not a-C0.

 

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DEPARTMENT OF MATHEMATICS

NAGARJUNA UNIVERSITY

NAGARJUNA NAGAR-522 510

INDIA

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