Fuzzy Semi Compact and Fuzzy Semi Coercive Mappings

Introduction The concept of fuzzy set and fuzzy set operations were first introduced by Zadeh [13] in 1965. Several other authors applied fuzzy sets to various branches of mathematics. One of these objects is a topological space. At the first time in 1968, Chang in [2] formulated the natural definition of fuzzy topology on a set and investigated how some of the basic ideas and theorems of point-set topology behave in generalized setting. Changʼs definition on a fuzzy topology is very similar to the general topology by exchange all subsets of a universal set by fuzzy subset but this definition is not investigate some properties if we comparison with the general topology. For example in a general topology, any constant mapping is continuous while this idea is not true in Changʼs definition on fuzzy topology. One of the very important concepts in fuzzy topology is the concept of mapping. There are several types of mapping. The purpose of this paper is to introduced and study the concept of fuzzy semi compact and fuzzy semi coercive mappings in fuzzy setting and explain the relation between them.


Introduction
The concept of fuzzy set and fuzzy set operations were first introduced by Zadeh [13] in 1965.Several other authors applied fuzzy sets to various branches of mathematics.One of these objects is a topological space.At the first time in 1968, Chang in [2] formulated the natural definition of fuzzy topology on a set and investigated how some of the basic ideas and theorems of point-set topology behave in generalized setting.Changʼs definition on a fuzzy topology is very similar to the general topology by exchange all subsets of a universal set by fuzzy subset but this definition is not investigate some properties if we comparison with the general topology.For example in a general topology, any constant mapping is continuous while this idea is not true in Changʼs definition on fuzzy topology.One of the very important concepts in fuzzy topology is the concept of mapping.There are several types of mapping.The purpose of this paper is to introduced and study the concept of fuzzy semi compact and fuzzy semi coercive mappings in fuzzy setting and explain the relation between them.§ 1. Preliminaries we present some fundamental definitions and propositions which are needed in the next sections.

Definition (1.1) [13]
Let be a non-empty set and let be the unit closed interval, i.e., =[ ].A fuzzy set in is a function from into the unit closed interval (i.e., be a function).
is interpreted as the degree of membership of element in a fuzzy set for each .A fuzzy set in can be represented by the set of pairs: {( ): }.The family of all fuzzy sets in is denoted by .

Remark (1.2)[7]
(a) The empty fuzzy set is a fuzzy set has membership defined by ,for all .The empty fuzzy set on a set denoted by .
(b) The universal fuzzy set is a fuzzy set has membership defined by ,for all .The universal fuzzy set on a set denoted by .

Definition (1.3)[2]
Let and be fuzzy sets in a set .

Remark (1.4) [5]
Let and be fuzzy sets in a set , then and .

Definition (1.6)[1]
Let and be two non-empty sets, be a mapping.For a fuzzy set in with membership mapping .Then the inverse image of under , written as , is a fuzzy set in whose membership mapping is defined by: for all .(i.e., ) Conversely, let be a fuzzy set in with membership mapping . The image of under written as , is a fuzzy set in whose membership function is given by: { for all , where { }.

Definition (1.7)[4,11]
A fuzzy point (singleton) in is a fuzzy set in defined by: ={ For each , the single point is called the support of and ] its value.We denote the class of all fuzzy points in by FP .A fuzzy point is said to be contained in a fuzzy set or belongs to , i.e., if and only if .
Two fuzzy points and in are said to be distinct if and only if their supports are distinct.

Definition (1.8)[9]
A fuzzy set in is called quasicoincident (in short q-coincident) with a fuzzy set in , denoted by if and only if , for some .Otherwise if , for every , then is called not q-coincident with and it is denoted by ̃ .
Lemma (1.9) [7] Let , and be fuzzy sets in , { }be a family of fuzzy sets in , and be a fuzzy point in .Then: Proposition (1.10) [4] Let be a fuzzy point in and be any fuzzy subset of .Then if and only if ̃ .
Definition (1.11) [2] Let be a set.A fuzzy topology on is a family of fuzzy sets in , which satisfies the following conditions: (1) ,

Definition (1.12) [6]
A fuzzy set in an fts is called a fuzzy quasi-neighborhood (in short qnhd) of a fuzzy point in if and only if there exists such that and .The family of all fuzzy q-nbds of is called the system of fuzzy q-nbds of and it is denoted by .

Definition (1.13) [8]
An fts is called fuzzy Hausdorff or fuzzy -space if and only if for any pair of distinct fuzzy points , in , there exists , such that = .

Definition (1.14)[7]
Let be a fuzzy set in and be a fuzzy topology on .Then the induced fuzzy topology on is the family of fuzzy subsets of which are the intersection with of fuzzy open set in .The induced fuzzy topology is denoted by , and the pair is called a fuzzy subspace of .
Definition (1.15) [7] Let be a fuzzy topological space and .Then : The union of all fuzzy open sets contained in is called the fuzzy interior of and denoted by .i.e. , .The intersection of all fuzzy closed sets containing is called the fuzzy closure of and denoted by .i.e. , [7] The interior of a fuzzy set is the largest open fuzzy set contained in and trivially , a fuzzy set is fuzzy open set if and only if = .

Remarks(1.16)
The closure of a fuzzy set is the smallest closed fuzzy set containing and trivially , a fuzzy set is fuzzy closed if and only if = .
Proposition (1.17) [12] Let be a fuzzy point in and be any fuzzy set in an fts , then if and only if for every fuzzy open set in such that is q-coincident with .

Definition (1.18) [2]
A mapping from an fts into an fts is called fuzzy continuous if and only if the inverse image of each fuzzy open set in is a fuzzy open set in .

Example (1.19)
Let be an fts, then the identity mapping is fuzzy continuous.

Proposition (1.20) [7]
Let and be fuzzy spaces, and be a mapping, then the following statements are equivalent:

Definition (1.23) [6]
A fuzzy set in an fts is called fuzzy semi-open if there exists a fuzzy open subset of such that .The complement of a fuzzy semi-open set is defined to be fuzzy semi-closed.

Proposition (1.24) [6]
A fuzzy subset in an fts is fuzzy semi-open if and only if .The converse of a and b of Remark (1.25), are not true in general as the following example.

Definition (1.27)[4]
Let be a fuzzy set of an fts .Then the fuzzy semi interior of , denoted by , is the union of all fuzzy semi open sets in which are contained in .It is evident that is fuzzy semi open if and only if .

Definition(1.28) [6]
Let be a fuzzy set of an fts .Then the fuzzy semi-closure of , denoted by , is the intersection of all fuzzy semi-closed subsets of containing .It is clear that is fuzzy semi-closed if and only if .

(d) .
Definition (1.30) [3] A fuzzy set in an fts is called a fuzzy semi quasi-neighborhood of a fuzzy point if there exists a fuzzy semi open set in such that .The family of all fuzzy semi q-nbds of is called the system of fuzzy semi q-nbds of and it is denoted by .

Remark (1.31)[3]
Every fuzzy q-nbd subset of an fts is a fuzzy semi q-nbd.

Definition (1.33) [10]
A mapping from an fts into an fts is said to be:

. Fuzzy Compact and Fuzzy Semi Compact Space
This section contains the definitions , proportions and theorems about fuzzy compact space and fuzzy semi compact space and we give a new results .

Definition (2.1) [6,8]
A mapping FP is called a fuzzy net in and is denoted by { }, where is a directed set.

If
for each , where , and ] , then the fuzzy net is denoted as { } or simply { }.

Definition (2.2) [6,8]
A fuzzy net { } in is called a fuzzy subnet of a fuzzy net { } if and only if there exists a mapping such that : 1. , that is, for each .

For each
there exists some such that, if with , then .
We shall denote a fuzzy subnet of a fuzzy net { } by { }.

Definition (2.3)[6]
Let be an fts and { } be a fuzzy net in and be a fuzzy set in .Then is said to be: Proposition (2.8) [5] An fts is fuzzy -space if and only if every converges fuzzy net on has a unique limit point.

Definition (2.9) [10]
Let be an fts, FP and let } be a fuzzy net in .Then: is called a semi cluster point of a fuzzy net , denoted by , if for each , is frequently with .
is said to be semi convergent to and is called a semi limit point of , denoted by , if for each , is eventually with .

Proposition (2.10) [10]
A fuzzy point issemi cluster point of a fuzzy net { } if and only if it has a fuzzy subnet which is semi convergent to .

Definition (2.13)
An fts is called fuzzy semi Hausdorff or fuzzy semi -space if and only if for any pair of distinct fuzzy points , in , there exists , such that .

Proposition (2.14)
An fts is fuzzy semi -space if and only if every semi converges fuzzy net on has a unique semi limit point.

Proof :
Let  [6] Let be a fuzzy set in an fts .Then is said to be fuzzy compact if for every fuzzy open cover of has a finite subcover of .Also, an fts is called fuzzy compact if every fuzzy open cover of has a finite sub cover.

Example (2.17)
The indiscrete fuzzy topological space is fuzzy compact.

Proposition (2.18) [5]
The fuzzy continuous image of a fuzzy compact set is fuzzy compact.Proposition (2.19) [7] Let be a fuzzy subspace of an fts , and be a fuzzy set in .Then is a fuzzy compact set in if and only if is a fuzzy compact set in .
Proposition (2.20) [2] An fts is fuzzy compact if and only if each family of fuzzy closed sets which has the finite intersection property has a nonempty intersection.

Corollary (2.21) [7]
A fuzzy closed subset of a fuzzy compact space is fuzzy compact.[7] A fuzzy filter base on is a nonempty subset of such that ( If then such that .

Definition(2.23) [10]
A fuzzy point in a fuzzy topological space is said to be a fuzzy semi-cluster point of a fuzzy filter base on if for all Proposition (2.24) [7] A fuzzy topological space is fuzzy compact if and only if every fuzzy filter base on has a fuzzy cluster point .Proposition (2.25) [7] An fts is fuzzy compact if and only if every fuzzy net in has a point.

Proposition (2.26) [7]
An fts is fuzzy compact if and only if each fuzzy net in has a -convergent fuzzy subnet.Proposition (2.27) [7] A fuzzy compact subset of a fuzzyspace is fuzzy closed.Proposition (2.28) [7] In any fts , the intersection of any fuzzy closed set with any fuzzy compact set is fuzzy compact.

Definition (2.29)[7]
Let be an fts.A fuzzy subset of is called fuzzy compactly closed if for every fuzzy compact set in , is fuzzy compact.

Example (2.30)
Every fuzzy subset of a fuzzy indiscrete space is fuzzy compactly closed.Proposition (2.31) [7] Let be a fuzzy -space.A fuzzy subset of is fuzzy compactly closed if and only if is fuzzy closed.

Definition (2.32) [10]
A family of fuzzy sets is called a fuzzy semi open cover if is a cover of and each member of is a fuzzy semi open set.A subcover of is a subfamily of which is also a cover of .

Definition (2.33) [10]
Let be a fuzzy set in an fts .Then is said to be fuzzy semi compact if for every fuzzy semi open cover of has a finite subcover of .Also, an fts is called fuzzy semi compact if every fuzzy semi open cover of has a finite sub cover.

Remarks(2.34)
Every fuzzy semi compact space is fuzzy compact.

Proposition (2.35)
An fts is fuzzy semi compact if and only if each family of fuzzy semi closed sets which has the finite intersection property has a non-empty intersection.

Proof :
Let { } be a family of fuzzy semi closed sets with the finite intersection property.Supposed that .Then .Since { } is a collection of fuzzy semi open sets cover of , then from the semi-compactness of it follows that there exists a finite subset such that .Then which gives a contradiction and therefore .
Let { } be a collection of fuzzy semi open sets cover of .Suppose that for every finite subset , we have .Then . Hence { } satisfies the finite intersection property.Then from hypothesis we have which implies and this contradicting that { } is a fuzzy semi open cover of .Then is fuzzy semi compact.

Proposition (2.36) [10]
A fuzzy topological space is fuzzy semi compact if and only if every fuzzy filter base on has a fuzzy semicluster point .Proposition (2.37) [10] An fts is fuzzy semi compact if and only if every fuzzy net in has a fuzzy semicluster point.[10] An fts is fuzzy semi compact if and only if each fuzzy net in has semi convergent fuzzy subnet.
(a) is fuzzy continuous.(b) The inverse image of each fuzzy closed set in is fuzzy closed in .(c) For each fuzzy set in , .(d) For each fuzzy set in , .(e) For each fuzzy set in , .Definition (1.22) [6] A mapping from an fts into an fts is called fuzzy closed (fuzzy open) if is a fuzzy closed (fuzzy open) set in for each fuzzy closed set (fuzzy open set) in .
(a) Every fuzzy open set is fuzzy semiopen.(b) Every fuzzy closed set is fuzzy semiclosed.
(a) fuzzy semi continuous if is a fuzzy semi open set in , for each fuzzy open set in .(b) fuzzy semi irresolute if is a fuzzy semi open set in ,for each fuzzy semi open set in .§ 2