The idea of a class of sets with a continuum of grade of membership, ranging between zero and one, was first introduced by Zadeh in 1965. A larger degree of membership of an object reflects a stronger sense of belonging to a set. If A is a set in the ordinary sense of the term, then its membership takes only two values, 0 and 1. The notions of inclusion, union, intersection, complement, relation and convexity can be extended to such sets [
Fuzzy closure spaces were introduced by [
and the isotone axiom
where 
is the closure operator on a nonempty set
.
is defined as a function
. Here 
represents the degree of membership of 
in the fuzzy set
.
of a set 
can be identified with its characteristic function 
defined by;
. Thus fuzzy sets generalize ordinary sets [
and 
be 
-valued functions defined on a fixed set, i.e fuzzy sets on
. Then according to [
implies 
for every
.
implies 
for every
.
.
.
. These are [0,1]-valued functions.
and 
in
;
and 
are equal if and only if
.
is contained in 
if and only if
.
and 
is
.
and 
is
.
is
.
be fuzzy sets. Then the union of 
is defined by;
is defined by;
are crisp, i.e. they are characteristic functions, then these suprema and infima are actually maxima and minima.
is a collection 
of fuzzy sets in 
satisfying
, where 
is equivalent to the empty set.
and 
belong to
, then
.
are fuzzy, then
. The members of 
are called open fuzzy sets.
is called a fuzzy topological space. Fuzzy sets of the form
, where 
is fuzzy are called closed fuzzy sets.
and 
be sets and 
be a function. For a fuzzy set 
in
, the inverse image of 
under 
is the fuzzy set 
in 
defined by;
. That is
.
in
, the image of 
under 
is the fuzzy set 
in 
defined for 
by;
and
, then according to [
is fuzzy continuous if the inverse image under 
of any open fuzzy set 
in 
is an open fuzzy set in
, i.e 
whenever
.
on a fuzzy topological space 
is fuzzy continuous.
be a fuzzy topological space. The closure 
and interior 
of a fuzzy set 
in 
are defined respectively by [
is the smallest closed fuzzy set larger than 
and that 
is the largest open fuzzy set smaller than
. These definitions coincide with their analogous definitions on ordinary sets.
be the collection of all mappings from 
to the unit interval
, i.e 
is the collection of all fuzzy sets on the non-empty set
. Then from [
is a fuzzy closure operator if and only if
constant.
for every
.
for every
.
for every
.
is closed if
. A fuzzy interior operator 
is the dual of a closure operator. It is defined by;
constant.
for every
.
for every
.
for every
.
is open if
.
is a function 
satisfying the following three axioms;
constant.
for every
.
for every
.
is called a fuzzy closure space or fcs. Clearly these axioms can easily be seen to be similar to the Kuratowski axioms in [
is a function 
satisfying the following two axioms;
constant.
,
.
is called a fuzzy isotone space.
, two fuzzy subsets 
and 
are called semi-separated if
.
and 
are separated if there exists open fuzzy sets
, V with 
and 
such that
. The openness of fuzzy sets on 
is defined using the dual of the closure operator, i.e the interior operator.
. Then 
and 
are semi-separated in 
if and only if 
and 
are semi-separated in
, where 
is the relativization of 
into
.
be a fuzzy isotone space and 
be semi-separated fuzzy sets in
. Then
.
hence
.
hence
and 
are semi-separated in
.
be a subspace of the fuzzy isotone space 
and 
and 
are semi-separated in
. Of course 
and
.
.
.
and 
are semi-separated in
.
is normal if for every nonempty pair of fuzzy sets 
and 
in 
such that 
there exists a fuzzyopen set 
such that 
and
.
be a normal fuzzy isotone space. Then for each pair of disjoint fuzzy subsets 
and Ф, there exists a fuzzy continuous function 
such that 
on 
and 
on
. Clearly, this characterization is analogous to the definition of normality via the existence of an Urysohn function on a normal topological space.
is said to be completely normal if every fuzzy subspace of 
is normal.
is completely normal if and only if for every pair 
and 
of fuzzy subsets with 
then there exists disjoint fuzzy sets 
and
.
be completely normal and 
be fuzzy sets with
. Denote
, a subspace of 
by
. Then 
since
.
.
and 
are closed sets in 
. Notice
.
is completely normal, then 
is normal and hence there exists 
and 
in 
such that 
and
.
be a subspace of 
and 
such that 
and 
with
. Then
. Similarly,
. Therefore, by the hypothesis of the theorem, there exists disjoint fuzzy sets
and
.
and 
are disjoint and contained in 
and
. Hence 
is normal and 
is therefore completely normal.
.
is called a 
-set if and only if
where
. A set 
is called an 
-set if
.
is perfectly normal if 
is normal and for every fuzzy subset 
of
, 
is a 
-set. That is for every closed fuzzy set 
in
,
. Equivalently, a normal isotonic space 
is perfectly normal if every open fuzysubset of 
is an 
-set. That is, for every fuzzy set
, then
.
is perfectly normal if for every 
such that
, 
and
, then 
a fuzzy continuous function 
that precisely separates 
and
. That is 
and 
is completely normal.
be perfectly normal fuzzy isotone space. Then for every closed fuzzy set 
there exists a fuzzy continuous function 
such that
.
be a subspace of 
and 
be closed. Then there exists a fuzzy closed set 
such that
. Since 
is closed in
, then there exists a fuzzy continuous function 
. But 
is also fuzzy continuous and
. Therefore 
is perfectly normal and hence normal. This implies that X is completely normal and hence perfect normality implies complete normality.