Abstract:

In this paper we introduce fuzzy ? –

generalized Baire Spaces, fuzzy weakly generalized Baire space, fuzzy

generalized ?-Baire space and discuss about some of

its properties with suitable examples.

Key

words:

Fuzzy

? – open sets, fuzzy ? – generalized open sets, fuzzy ? – generalized Baire

space, fuzzy weakly generalized Baire space and fuzzy generalized ?-Baire space.

Introduction:

The theory of fuzzy sets was initiated by L.A.Zadeh in his

classical paper 9 in the year 1965 as an attempt to develop a mathematically precise

framework in which to treat systems or phenomena which cannot themselves be

characterized precisely. The potential of fuzzy notion was realized by the

researchers and has successfully been applied for investigations in all the

branches of Science and Technology. The paper of C.L.Chang 2 in 1968 paved

the way for the subsequent tremendous growth of the numerous fuzzy topological

concepts.

The concepts of ?-generalized closed sets have been studied

in classical topology in 3. In this paper we introduce the fuzzy ?–generalized,fuzzy

weakly generalized and fuzzy generalized ? – nowhere dense sets and fuzzy ?-generalized,

fuzzy weakly generalized and fuzzy generalized

? –Baire spaces with suitable examples.

Preliminaries:

Now review of

some basic notions and results used in the sequel. In this work by (X,T) or

simply by X, we will denote a fuzzy topological space due to Chang 2.

Definition 2.1 1 Let ? and ? be any

two fuzzy sets in a fuzzy topological space (X, T). Then we define:

???

: X ? 0,1 as follows: ??? (x) = max {?(x), ?(x)};

???

: X ? 0,1 as follows: ??? (x) = min { ?(x), ?(x)};

?

=

? ?(x) = 1-

?(x).

For a family

? I of fuzzy sets in (X, T), the union ? =

and intersection

? =

are

defined respectively as

, and

.

Definition 2.2 2

Let (X,T) be a fuzzy topological space. For a fuzzy set ? of

X, the interior and the closure of ? are defined respectively as

and cl

.

Definition 2.2 3

Let (X,T) be a topological space. For a fuzzy set ? of X is a

? – generalized closed set (briefly ?g-closed) if ?cl(?) ? µ whenever ? ? µ and

µ is fuzzy open in X.

Definition 2.3 8

A fuzzy set ? in a fuzzy topological space (X,T) is called

fuzzy dense if there exists no fuzzy closed set ? ? (X,T) such that ?