Fuzzy Semi Pre Homeomorphism in Fuzzy Topological Spaces

The aim of this paper is to introduce and study new class of fuzzy function called fuzzy semi pre homeomorphism in a fuzzy topological space by utilizing fuzzy semi pre-open sets. Therefore, some of their characterization has been proved; In addition to that we define, study and develop corresponding to new class of fuzzy semi pre homeomorphism in fuzzy topological spaces using this new class of functions. Keyword: Fuzzy topological spaces, fuzzy semi pre-open sets, fuzzy semi pre homeomorphism function and fuzzy semi pre* homeomorphism function.


Introduction
Zadeh in [1] introduced the fundamental concept of fuzzy sets and fuzzy set operations in his standard paper. Thereafter, some researchers have applied some different basic notions from general topology by utilizing the fundamental idea of fuzzy sets and was subsequently expanded by developing important theories of fuzzy topological spaces. The notion of family in fuzzy sets naturally plays a very significant role in the study of the recent concept of fuzzy topology introduced by Chang and investigated some of their properties. This paper is to introduce and investigate some new types of fuzzy semi pre continuous functions and fuzzy semi pre homeomorphism functions via fuzzy semi pre-open sets. Also, the relationships between these functions and other types are discussed. Several properties of these new notions are investigated and the connections between them are studied. Moreover, some results in this topic with some properties and corollaries are developed.

Preliminaries
Throughout this paper X denotes anon empty set and I denote the interval [0, 1]. A fuzzy set in X is a function from X in to I. The value represents the degree of membership of ∈ in the fuzzy set A. The family of all fuzzy sets on a crisp set X will be denoted by A family ̃ of fuzzy sets in X is called a fuzzy topology for X iff (1)  A fuzzy set in a crisp set X is called a fuzzy point if it takes the grade of membership (0) for all ∈ 1 , say, ∈ . If its grade of membership at is ( 0 1 ), we denote this fuzzy point by , where the point is called its support [6]. For any fuzzy and any fuzzy set , we write ∈ iff .

Definition 2.1 [7]:
Let : ,̃  , , let be a fuzzy set in Y with membership function . Then the inverse of written as is a fuzzy set in X whose membership function is defined by ∈ .
Conversely, let be a fuzzy set in X with membership function . The image of written as is a fuzzy set in Y whose membership function is given by

, ℎ
For each y belong to Y, where .
(2) For any fuzzy set in X, 1 .
(3) For any fuzzy set in X, and if is one-to-one.
(4) For any fuzzy set in Y, and if is onto.

Fuzzy Semi Pre Open Sets
This section is devoted to definitions, and some theorems of fuzzy semi pre-open set and fuzzy semi pre closed set. Moreover, we study in this section some remarks and relationship between fuzzy semi pre-open (semi pre closed) set and fuzzy open (closed) set. Also some of their properties which we need them in our study are discussed.

Definition 3.1 [11]:
Let (X,) be a f.t.s, a fuzzy set in X is called:

Notation 3.2:
In f.t.s (X,), we denote: (1) The family of all fuzzy semi pre-open sets of X by F. S. P. O. X .
(2) The family of all fuzzy semi pre closed sets of X by F. S. P. C. X .

Remark 3.4:
Every fuzzy open (resp. closed) set is fuzzy semi pre-open (resp. semi pre closed) set. But the converse is not true in general.

Example 3.5:
Let , and let , are fuzzy sets of X defined as follows: , be a f.t.s. on X, it's clear that: (1) The fuzzy set is fuzzy semi pre-open set in X but it is not a fuzzy open set.
(2) The fuzzy set is fuzzy semi pre closed set but it is not a fuzzy closed set.

Remark 3.7 [5]:
(1) Let and be two fuzzy semi pre-open sets then , need not to be fuzzy semi pre open set.
(2) Let and be two fuzzy semi pre closed sets then , need not to be fuzzy semi pre closed set.

Let
, and let , are fuzzy sets of defined as follows: (2) The fuzzy set and are fuzzy semi pre closed sets in X, but , is not a fuzzy semi pre closed set.

Fuzzy Semi Pre Homeomorphism in fuzzy Topological Spaces.
In this section we introduce the fuzzy semi pre homeomorphism functions and fuzzy semi pre* homeomorphism functions in fuzzy topological spaces. Some of their properties and characterization have been proved and discussed in details.

Theorem 4.4:
Let : ,̃  , be bijection fuzzy function then the below statements are equivalent.
(1) f is a fuzzy semi pre-open function.
(2) f is a fuzzy semi pre closed function.
(3) f -1 is a fuzzy semi pre continuous function.

Proof:
(1) → (2) Let be fuzzy closed set belong to X, then 1 is a fuzzy open set belong to X since f is a fuzzy semi pre-open function. Let : ,̃  , be a bijective fuzzy function, then the below statements are equivalent.
(1) f is a fuzzy semi pre continuous and a fuzzy semi pre-open functions.
(2) f is a fuzzy semi pre continuous and a fuzzy semi pre closed functions.
(3) f is a fuzzy semi pre homeomorphism function. Proof: (1) → (2) Follows from proof Theorem Let : ,̃  , be bijection fuzzy function, f is called a fuzzy semi pre  homeomorphism iff f and f -1 are fuzzy semi pre irresolute functions (abbreviated as . . * . ℎ. . ).We say the spaces (X,) and (Y, ) are fuzzy semi pre  homeomorphism if there exist a fuzzy semi pre  homeomorphism from (X,) on to (Y, ). The family of fuzzy semi pre  homeomorphism from a f.t.s (X,) to itself is denoted by fuzzy semi pre  Homeomorphism(X,) (abbreviated as . . * . ℎ.(X,)).

Theorem 4.7
Every fuzzy semi pre  homeomorphism function is a fuzzy semi pre homeomorphism function.

Proof:
Let : ,̃  , be a fuzzy semi pre  homeomorphism function, So, f and f -1 are fuzzy semi pre irresolute functions. Hence, by Proposition (3.11) f and f -1 are fuzzy semi pre continuous functions. Thus, is a fuzzy semi pre homeomorphism function (1) f is a fuzzy semi pre irresolute and a fuzzy semi pre  -open functions.
(2) f is a fuzzy semi pre irresolute and a fuzzy semi pre  closed functions.

Conclusions
In this paper, we been developed the new types of fuzzy homeomorphism function in fuzzy topological spaces called fuzzy semi pre homeomorphism function and fuzzy semi pre* homeomorphism function. In addition to that we developed the relationship between these new types of fuzzy homeomorphism function in fuzzy topological spaces with some new kinds of fuzzy semi pre continuous function. Finally, we structure a group under the usual composition functions by using these new types of fuzzy homeomorphism function. As follows: (I) Every fuzzy semi pre  homeomorphism function is a fuzzy semi pre homeomorphism function.
(II) Let : ,̃  , be a bijective fuzzy function, then the below statements are equivalent.
(1) f is a fuzzy semi pre irresolute and a fuzzy semi pre  -open functions.
(2) f is a fuzzy semi pre irresolute and a fuzzy semi pre  closed functions.
(3) f is a fuzzy semi pre  homeomorphism function. (III) Let ,̃ be the set of all fuzzy semi pre  homeomorphism (or . . * . ℎ. ,̃ for short), then ,̃ is a group under the usual composition functions.