# Abstract: topological concepts. The concepts of ?-generalized

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.

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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 ?