GOOD EXTENSIONS FOR SOME (FUZZY) TOPOLOGICAL PROPERTIES TO (FUZZY) CLOSURE SPACES

Abstract

A closure operator on a set X is a function c: P(X) P(X) which satisfies various combinations of the following axioms:

Cl: c (<j>) = <j>.

C2: c (A) ::J A.

C3: c (Au B)= c (A) u c (B).

C4: Ac B => c (A)c c (B).

C5: c (c (A))c c (A).

There are many possible interpretations of a closure operator. For example:

1) If c (A)= n IF: FE :F, F ::J A), where :Tis a family of sets, then an operator so defined must satisfy C2, C4 and C5.

2) If c (A)= Au {x : xis a cluster point of A), then c must satisfy C2 and other axioms, depend on the properties of the cluster points of the set A, like Cl and C3.

3) If (A n B t= <j> => A IS near B) and c (A) is defined as c (A) =

{x: {x) is near A), then c must satisfy Cl, C2 and C3.

v In [3], E. Cech has studied a basic proximity structure on a set X t= <j>.

The closure operator induced by such a structure is, in general, not a

Kuratowski Closure Operator. It may fail to satisfy C5. However, it