PAIRWISE CO AND PAIRWISE C1 SPACES

N.Palaniappan and S.Pious Missier

(Department of Mathematics, Alagappa University, karaikudi-630 003)

ABSTRACT: The aim of this paper is to introduce the concepts of pairwise C0, pairwise C1, pairwise semi-C0 and pairwise semi-C1 bitopological spaces and study some of their basic properties. The implications of these new separation axioms among themselves and with well-known axioms pairwise semi-T0 and pairwise semi-T1 are obtained.

KEYWORDS: Pairwise C0, pairwise C1,pairwise a-C0, pairwise a-C1 Semi-C0 and pairwise semi-C1 spaces.

1. INTRODUCTION

The concept of bitopological spaces was first introduced by J.C.Kelly. [1](1963). N. Levine 1963[2] defined the notion of semi- open set in topological spaces. Olas Njastad[4] introduced the notion of a-open set in topological spaces. S.N. Maheswari et al[3] defined pairwise semi-T0 and pairwise semi-T1 in bitopological spaces. Recently M.K.R.S. Veera Kumar[7] has defined and characterized the new separation axioms C0, C1, a-C0 and a-C1 in topological spaces. In the present paper, we generalize the notions of C0, C1, a-C0 and a-C1 spaces to bitopological spaces. Throughout the present paper (X,t1,t2) always represent bitopological spaces on which no separation axioms are assumed unless otherwise explicitly defined. For any subset A, ti-Int(A) and ti-Cl(A) denote the interior of A and the closure of A with respect to ti (i=1,2).

2. PRELIMINARIES. Let (X, t) be a topological space and AÌX. A is said to be an a-open [5] (res.semi-open[2]) if AÍInt(Cl(Int(A))) (resp. AÍCl(Int(A))). The family of a-open set (res. semi-open) sets in a space X is denoted by aO(X,t) (resp. SO(X,t)). The complement of an a-open (semi-open) is called a-closed (semi-closed). The intersection of all a-closed sets (resp.semi-closed sets) containing A is called a-closure (semi-closure) of A and is denoted by acl(A)(res.scl(A)).

DEFINITION 2.1[7]. A space (X,t) is called

(a)  C0 (semi-C0) if x,y ÎX, x¹y, there exists G Ît(SO(X,t)) such that cl(G) (scl(G)) contains only one of x and y but not the other.

(b)  C1 (semi-C1) if, for x,y ÎX, there exist G, HÎt(SO(X,t)) such that

xÎcl(G)(scl(G)), yÎCl(H)(scl(H)) but xÏCl(H) (scl(H)) and yÏCl(G)(scl(G)).

DEFINITION 2.2[6]. A bitopological space (X,t1,t2) is called a pairwise T0, if for any pair of distinct points of X, there a set which is either t1-open or t2-open containing one of the points but not the other.

DEFINITION 2.3 [3]. A bitopological space (X,t1,t2) is pairwise semi- T0 if for each pair of distinct points of X, there is a set which is either t1-semi-open or t2-semi- open containing one of the points but not the other.

DEFINITION 2.4[6]. A bitopological space (X,t1,t2) is pairwise T1 if for each pair of distinct points x and y of X there is a t1-open set U and a t2-openset V such that xÎU, yÏU and yÎV, xÏV.

DEFINITION 2.5[3]. A bitopological space (X,t1,t2) is pairwise semi- T1 if for each pair of distinct points x,y of X, there exists a t1-semi- open set U and t2-semi-open set V such that x ÎU, yÏU and yÎV, xÏV.

3. Pairwise C0 and Pairewise semi-C0 bitopological spaces.

DEFINITION 3.1. A bitopological space (X,t1,t2) is said to be pairwise C0 if for each pair of distinct points of X, there is a set which is either t1-open or t2-open whose closure contains one of the points but not the other.

EXAMPLE 3.2. Let X={a,b,c},t1={X,f,{a},{b},{a,b}} and t2={X,f,{b},{a,b},{b,c}}. Then (X,t1,t2) is pairwise C0

PROPOSITION 3.3. If (X,t1,t2) is pairwise C0 space, then it pairwise T0.

PROOF . Easy and hence Omitted.

REMARK 3.4. The converse of the Proposition 3.3 need not be true as can be seen from the following example.

EXAMPLE 3.5. Let X={a,b,c}, t1={X,f,{a},{b,c}} and t2={X,f,{b}}. Then (X,t1,t2) is pairwise T0 but not pairwise C0.

PROPOSITION 3.6. A space (X,t1,t2) is pairwise C0 if either (X,t1) or (X,t2) is C0.

PROOF. Easy and omitted.

REMARK 3.7. The converse of the Proposition 3.6 need not be true as can be seen from the following example.

EXAMPLE 3.8. Let X={a,b,c}, t1={X,f,{a},{b,c}} and t2={X,f,{b},{a,c}}. Then (X,t1,t2) is pairwise C0 but neither (X,t1) nor (X,t2) is C0

DEFINITION 3.9. A bitopological space (X,t1,t2) is said to the pairwise semi-C0 if for each pair of distinct points of X, there exists a set which is either t1-semi-open or t2-semi- open whose semi-closure containing one of the points but not the other.

REMARK 3.10. It is evident that every pairwise C0 is pairwise semi- C0.

THEOREM 3.11. If (X,t1,t2) is pairwise semi-C0 then it is pairwise semi-T0.

PROOF. Suppose that (X,t1,t2) is pairwise semi-C0. Let x,y ÎX, x¹y.

Let U be a t1-semi-open set such that xÎt1-scl(U) but yÏt1-scl(U).

Then (X-t1-scl(U)) is a t1-semi-open set containing y but not x. In the similar way if xÏt1-scl(U) and yÎt1-scl(U), then the t1-semi- open set (X-t1-scl(U)) containing x but not y. Thus (X,t1,t2) is pairwise semi-T0 space.

REMARK 3.12. The converse of the Theorem 3.12 need not be true, as can be seen by the following example.

EXAMPLE 3.13. Let X={a,b,c} t1={X,f,{a},{b,c}} and t2={X,f,{b}}. Then (X,t1,t2) is pairwise semi- T0 but not pairwise semi- C0.

4. Pairwise C1 and Pairwise semi-C1 Spaces.

DEFINITION 4.1. A bitopological space (X,t1,t2) is pairwise C1 if for each pair of distinct points x,y of X, there exists a t1-open set U and t2-open set V such that xÎt1 - Cl(U), yÎt2-Cl(V) but xÏt2 - Cl(V) and yÏt1 - Cl(U).

PROPOSITION 4.2. If (X,t1,t2) is pairwise C1 space then it is pairwise C0

PROOF. Evident.

REMARK 4.3. The converse of the Proposition 4.2 need not be true as can be seen by the following example.

EXAMPLE 4.4. Let X={a,b,c}, t1={X,f,{a},{b},{a,b}} and t2={X,f,{b},{a,b},{b,c}}. Then (X,t1,t2) is pairwise C0 but not pairwise C1.

DEFINITION 4.5. A bitopologiocal space (X,t1,t2) is pairwise semi-C1 if for each pair of distinct points x,y of X, there exists a t1-semi- open set U and a t2-semi- open set V such that xÎt1 - scl(U),yÏt1 - scl(U) and yÎt2 - scl(V),xÏt2 - Scl(V).

REMARK.4.6. It is evident that pairwise semi-C1 space is pairwise semi-C0. The following example shows that the converse need not be true.

EXAMPLE 4.7. Let X={a,b,c}, t1={X,f,{a},{b}{a,b}} and t2 ={X,f,{a},{b,c}}. Then (X,t1,t2) is pairwise semi-C0 but not pairwise semi-C1.

THEOREM 4.8. If (X,t1,t2) is pairwise semi-C1 then it is pairwise semi-T1.

PROOF. Suppose that (X,t1,t2) is Pairwise semi-C1 space.

Let x,yÎX and x¹y, then there exist a t1-semi-open set U and a

t2-semi- open set V such that xÎt1 - scl(U),yÏt1 - scl(U) and yÎt2 - scl(V), xÏt2 - scl(V).

Now yÎt1 - scl{y}Í (X-t1 - scl(U)) and xÏ(X-t1 -scl(U)). Then (X-t1 - scl(U)) is a t1-semi-open set containing y but not x. In the similar way, we can claim that there is a t2-semi-open set containing x but not y. Hence (X,t1,t2) is pairwise semi-T1

REMARK 4.9. The converse of the Theorem 4.8 need not be true as can bee seen from the following example.

EXAMPLE 4.10. Let X={a,b,c},t1={X,f,{a},{c},{a,c}} and t2={X,f,{b}}. Then (X,t1,t2) is pairwise semi-T1 but not pairwise semi-C1

5. Pairwise a-C0 and Pairwise a-C1 Spaces.

DEFINITION 5.1.[7] A topological space (X,t) is called a-C0 if, for each x,yÎX,x¹y there exists GÎaO(X, t) such that acl(G) contains only one of x and y but not the other.

DEFINITION 5.2[7]. A topological space (X,t) is called a-C1 if, for each x,yÎX,x¹y there exists G,HÎaO(X, t) such that xÎacl(G) yÎacl(H) but yÏacl(G) and xÏacl(H).

DEFINITION 5.3. A bitopological space (X,t1,t2) is pairwise a-C0 if for each pair x,y of distinct points in X there exists a set which is either a t1 - a- open set or a t2 - a - open set where a - closure cntaing one of the points but not the other.

DEFINITION 5.4. A bitopological space (X,t1,t2) is pairwise a-C1 if for each pair x,y of distinct points in X,there exists a t1-a-open set U such that xÎt1-acl(U)

but yÏt1-acl(U) and a t2-a-open set V such that yÎt2-acl(V) but xÏt2-acl(V).

PROPOSITION 5.5. Every pairwise a-C1 space pairwise a-C0

PROOF. Obvious.

REMARK 5.6. The following example shows that the converse of the Proposition 5.5 need not be true.

EXAMPLE 5.7. Let X={a,b,c}, t1={X,f,{a},{b},{a,b}} and t2={X,f,{b},{a,b},{b,c}}. Then (X,t1,t2) is pairwise a-C0 but not pairwise a- C1.

DEFINITION 5.8. Every pairwise a-C0 space is pairwise semi-C0.

PROOF. Obvious.

REMARK 5.9. The following example shows that the converse of the Proposition 5.8 need not be true.

EXAMPLE 5.10. Let X={a,b,c}, t1={X,f,{a},{b,c}} and t2={X,f,{a},{b},{a,b}}. Then (X,t1,t2) is pairwise semi - C0 but not pairwise a-C0.

PROPOSITION 5.11. Every pairwise a-C1 space is pairwise semi-C1.

PROOF. Obvious.

REMARK 5.12. The converse of the Proposition 5.11 need not be true as can be seen from the following example.

EXAMPLE 5.13. Let X={a,b,c}, t1={X,f,{a},{b},{a,b}} and t2={X,f,{b},{c},{b,c}}. Then (X,t1,t2) is pairwise semi - C1 but not pairwise a-C1.

PROPOSITION 5.14. Every pairwise a-C0 space is pairwise semi - T0

PROOF. By Proposition 5.8, we have, every pairwise a-C0 space is pairwise semi-C0, But by theorem 3.11, we have every pairwise semi-C0 space is pairwise semi-T0. Therefore every pairwise a-C0 space is pairwise semi-T0.

REMARK 5.15. The converse of the Proposition 5.14 need not be true. Consider the following example.

EXAMPLE 5.16. Let X={a,b,c}, t1={X,f,{a},{a,b}} and t2={X,f,{b}}. Then (X,t1,t2) is pairwise semi-T0 but not pairwise a-C0.

PROPOSITION 5.17. If the space (X,t1,t2) is pairwise a - C1, then it is pairwise

semi-T1.

PROOF. By Proposition 5.11, we have, every pairwise a-C1 space is pairwise semi-C1, But by Theorem 4.8, we have every pairwise semi-C1 space is pairwise semi-T1. Hence every pairwise a-C1 space is pairwise semi-T1.

REMARK 5.18. The converse of the Proposition 5.17 need not be true as can be seen from the following example.

EXAMPLE 5.19. Let X={a,b,c}, t1={X,f,{a},{c},{a,c}} and t2={X,f,{b}}. Then (X,t1,t2) is pairwise semi-T1 but not pairwise a-C1.

THE FOLLOWING DIAGRAM SHOWS SOME RELATIONSHIPS BETWEEN SEPARATION AXIOMS CONSIDERED IN THIS PAPER. A ® B, A ® B REPRESENT A IMPLES B AND A NEED NOT IMPLY B RESPECTIVELY. “PAIR” STANDS FOR “PAIRWISE”.

References

[1] J.C. Kelly, Biotopological space, Proc. London. Math. Soc. (13),1963,71-79

[2] N. Levin, Semi-open sets and semi-continuity in topological spaces,

Amer: Math. Monthly(70),1963,36-41

[3] S.N. Maheswari and R. Prasad, Some new separation axioms in

bitopological spaces, Mathemathuku BeCHNK, 12(27),1975, CTP,159-162.

[4] M.G. Murdeshwar and S.A. Naimpally, Quasi uniform topological spaces, Noordhoff, Croningen,1966.

[5] O. Njastad,On some classes of nearly open se”, Pacific.J.Math.(15),1965,961.

[6] I.L. Reilly, On bitopological separation properties, NantaMath.(2) ,1970,14-25.

[7] M.K.R.S. Veera Kumar, Some separation properties using a - open sets, Indian J. Math. V(40)(2),1998,143-147.

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