Study about fuzzyω-paracompact space in fuzzy topological space

The purpose of this paper is to introduce a new class of fuzzy paracompact space is named fuzzy ω- paracompact space on fuzzy topological space also study the relationships with fuzzy ω- separation axioms and we give some characterization on fuzzy ω- paracompact space by using fuzzy countable set also we study the fuzzy ω- paracompact subspace and consider some relationship between fuzzy paracompact space and fuzzy ω- paracompact space by using certain types of fuzzy ω- continuous functions.


Introduction
The concept, which we will be considered in this paper, is the so called "fuzzy sets" which is totally different from the classical concept which is called "a crisp set". The recent concept is introduced by Zadeh in 1965 [15], in which he defines fuzzy sets as a class of objects with a continuum of grades of membership and such a set is characterized by a membership function that assigns to each object a grade of membership ranging between zero and one, In (1968) Chang [2] introduced the definition of fuzzy topological spaces and extended in a straight forward manner some concepts of crisp topological spaces to fuzzy topological spaces. Later Lowen [10] (1976) redefined what is now known as stratified fuzzy topology.While Wong [13] in 1974 discussed and generalized some properties of fuzzy topological spaces. The note on paracompact space has been introduced by Ernest Michael [4] in (1953). Qutaiba Ead Hassanin [9 ] in (2005) introduced characterizations of fuzzy paracompactness. In this paper we introduce the concepts of fuzzy -open set and fuzzy -paracompact space and fuzzy -paracompact subspace on fuzzy topological space, and studied the relationships with fuzzyseparation axioms also we presented some types of fuzzy -continuous function and we give some characterization. And we obtained several properties. [15] et be a non empty set, and let be the unit inter al i.e , , a fuzzy set in is a function from into the unit inter al , , be a function fuzzy set in can be represented by the set of pairs:

Definition
, ̃ ): x X} the family of all fuzzy sets in X is denoted by I X .

Definition [6]
A fuzzy point is a fuzzy set such that : = r 0 if x = y , y X and = 0 if x y , y X, The family of all fuzzy points of will be denoted by

Definition [13]
A fuzzy point is said to belong to a fuzzy set in (denoted by : if and only if ≤ ̃

Proposition[13]
Let and be two fuzzy sets in with membership functions and respectively, then for all x X: -1.

Definition [7]
The support of a fuzzy set ̃, Supp ( ̃) , is the crisp set of all x X, such that ̃ > 0.

Definition [2]
A fuzzy topology is a family ̃ of fuzzy subsets in X, satisfying the following conditions:

Definition [8]
If ,T ,the complement of referred to 1 X denoted by , is defined by = 1 X -

Definition [1]
An fuzzy open set ̃ in a fuzzy topological space ,T is said to be clopen if its complement 1 X -̃ is an fuzzy open.

Definition [2]
A fuzzy set in a fuzzy topological space ( ,T is said to be a fuzzy nei hborhood of a fuzzy point in if there is a fuzzy open set in such that , X.

Definition [3]
Let ,T be a fuzzy topological space a family ̃ of fuzzy sets is open cover of a fuzzy set ̃ if and only if ̃  { G : G  ̃} and each member of ̃ is a fuzzy open set.

Definition [12]
Let B = { ̃ : } , C = { ̃ : } ( ) be any two collection of fuzzy sets in ,T , then C is a refinement of B if for each there exist such that ̃ ̃ .

Definition [5]
A fuzzy topological space ,T is said to be fuzzy connected, if it has no proper fuzzy clopen set. Otherwise it is called fuzzy disconnected.

Definition [15]
Let be a function from universal set X to universal set Y. Let ̃ be a fuzzy subset in 1 Y with membership function ̃ . Then, the inverse of ̃, written as ̃) , is a fuzzy subset of 1 X whose membership function is defined by ), for all x in X.If ̃ be a fuzzy subset in 1 X with membership function ̃ . The image of ̃, written as ̃) , is a fuzzy subset in 1 Y whose membership function is defined by From the above it is clear that:

Definition [14]
fuzzy set in a fuzzy topolo ical space , ̃ is called a fuzzy uncountable if and only if supp is an uncountable subset of X

Definition
Let be a fuzzy set in a fuzzy topological space ,T then, The -closure of is denoted by ω-cl( and defined by ω-cl ) = ⋂ { is a fuzzy ω-closed set in 1 X , ⊆ }

Definition
A fuzzy topological space (X, ̃) is called a fuzzy anti-locally-countable if each nonempty fuzzy open subset of 1 X is uncountable.

Definition
A fuzzy topological space (X, ̃) is said to be 1-ω-̃ if for each pair of distinct fuzzy point and of 1 X there exist fuzzy ω-open set ̃ such that either ̃ and ̃ or ̃ and ̃.
2-ω-̃ if for each pair of distinct fuzzy point and of 1 X there exist fuzzy ω-open sets ̃ and ̃ such that ̃ and ̃ and ̃ and ̃. 3-ω-̃ if for each pair of distinct fuzzy point and of 1 X there exist disjoint fuzzy ω-open sets ̃ and ̃ containing and respectively .

Definition
A fuzzy topological space (X, ̃) is called a fuzzy ω-regular space if for each fuzzy ω-closed subset of 1 X and a fuzzy point in 1 X such that , there e ist dis oint fuzzy ω-open sets ̃ and ̃ containing and respectively

Definition
A fuzzy topological space (X, ̃) is called a fuzzy ω-Normal space if for each pair of disjoint fuzzy ωclosed sets ̃ and in 1 X there exist disjoint fuzzy ω-open sets ̃ and ̃ containing ̃ and respectively

Theorem
A fuzzy topological space (X, ̃) is fuzzy ω-Normal if for each pair of fuzzy ω-open sets ̃ and ̃ in 1 X such that 1 X = ̃ ⋃ ̃ there are fuzzy ω-closed sets ̃ and ̃ contained in ̃ and ̃ respectively such that 1 X = ̃ ⋃ ̃ Proof: Obvious

Theorem
Every fuzzy ω-closed subspace of fuzzy ω-Normal space is fuzzy ω-Normal space. Proof: Obvious

Proposition
Every fuzzy ω-regular space is fuzzy ω-̃ space Proof : Let and be pair of fuzzy distinct points in a fuzzy ω-regular space 1 X ,Then is a fuzzy point of 1 X which is not in the fuzzy ω-closed subset { } of 1 X so by fuzzy ω-regularity of 1 X there exist fuzzy disjoint ω-open sets ̃ and ̃ containing and respectively, Hence 1 X is fuzzy ω-̃ space 

Definition
Two fuzzy families { ̃ } and { ̃ } of subset of a fuzzy space 1 X are said to be similar if for e ery finite subset Δ of Λ the fuzzy sets A   and B   are either empty or nonempty.

Definition
Let (X, ̃) be a fuzzy topological space a family W of fuzzy sets is -open cover of a fuzzy set A if and only if A  { G : G W} and each member of W is a fuzzy -open set. A sub cover of W is a sub family which is also cover.

Propositions
Let ,T be a fuzzy ω-paracompact space and let ̃ be a fuzzy subset of 1 X and ̃ be an fuzzy ω-

Propositions
Each fuzzy ω-paracompact fuzzy ω-regular (resp. fuzzy ω-̃ ) space is fuzzy ω-Normal space. Proof: Let ,T be an fuzzy ω-paracompact ω-̃ space and let be any fuzzy point in 1 X which is not in an arbitrary fuzzy ω-closed set ̃ of 1 X therefore for each ̃ there are disjoint fuzzy ωopen sets ̃ and ̃ containing and { } respectively so by (4.5 Propositions) there exist disjoint fuzzy ω-open sets ̃ and ̃ containing ̃ and respectively this shows that ,T is fuzzy ω-regular space, thus we have ,T fuzzy ω-paracompact fuzzy ω-regular. Let ̃ and ̃ be any fuzzy two disjoint fuzzy ω-closed subset of 1 X , since ̃ is fuzzy ω-closed so by fuzzy ω-regularity of 1 X for each ̃ there exist disjoint fuzzy ω-open sets ̃ and ̃ containing and ̃ respectively therefore By (4.5 Propositions) there exist disjoint fuzzy ω-open sets ̃ and ̃ containing ̃ and ̃ this showed that ,T is fuzzy ω-Normal space

Corollary
Every fuzzy ω-paracompact ̃ space is an fuzzy ω-Normal space. Proof: Follows by the fact(Every fuzzy ̃ space is an fuzzy ω-̃ space) and (4.5 Propositions)

.24 Corollary and 3.25 Theorem
,T is fuzzy paracompact

Proposition
If a fuzzy topological space ,T is fuzzy ω-̃ space and has a fuzzy subset ̃ which is fuzzy ωparacompact subset to 1 X then for each 1 X -̃ there e ist two dis oint fuzzy ω-open sets of 1 X containing and ̃ Proof: Let ̃ be fuzzy ω-paracompact subset of fuzzy ω-̃ space ,T and let be any fuzzy point of 1 X -̃ then for each ̃ there e ist fuzzy ω-open sets ̃ and ̃ such that ̃ and ̃ and ̃ ⋂ ̃ = this implies that ω-cl( ̃ ) ⋂ ̃ = hence ω-cl( ̃ ) for each ̃.

Proposition
If a fuzzy topological space ,T is fuzzy ω-regular space and ̃ is fuzzy subset of 1 X which is fuzzy ω-paracompact subset of 1 X then for each fuzzy ω-open set ̃ containing ̃ there e ist fuzzy ω-closed set ̃ containing ̃ and it is contained in ̃ furthermore ̃ is is fuzzy ω-Normal subspace of 1 X . Proof: Since a fuzzy topological space ,T is fuzzy ω-regular space so by (3.22 Proposition) and

Theorem
Let ,T be a fuzzy ω-disconnected space then the statements are equivalent: 1-,T is fuzzy ω-paracompact space 2-E ery fuzzy proper ω-closed subset of 1 X is fuzzy ω-paracompact subset of 1 X 3-E ery fuzzy proper ω-closed subset of 1 X is fuzzy ω-paracompact subspace 4-E ery fuzzy proper ω-clopen subset of 1 X is fuzzy ω-paracompact 5-There e ist a fuzzy proper ω-clopen subset ̃ of 1 X such that both ̃ and 1 X -̃ are fuzzy ωparacompact. ̃ ⋂ ̃ } Covering ̃ and 1 X -̃ respectively such that ̃ is fuzzy ω-openset in ̃ for each and ̃ is fuzzy ω-open set in 1 X -̃ for each , then both ̃ and ̃ are fuzzy ω-open sets in 1 X for each and Therefore { ̃ } is fuzzy ω-locally finite ω-open refinement of { ̃ } which covers 1 X , hence ,T is fuzzy ω-paracompact space Remark In the above theorem if ,T is fuzzy ω-connected space then the only fuzzy ω-clopen subset of 1 X are fuzzy empty set and 1 X itself so the condition that ,T is fuzzy ω-disconnected space is essential.

Proposition
Let ̃ be a fuzzy ω-clopen subset of a fuzzy topological space ,T then ̃ is fuzzy ω-paracompact subset if and only if ̃ is fuzzy ω-paracompact subspace. Proof: In view of (5.

Proposition
Let ̃ and ̃ be two fuzzy subset of a fuzzy topological space ,T if ̃ is fuzzy ω-closed and ̃ is fuzzy ω-paracompact subset to 1 X then ̃ ∩ ̃ is fuzzy ω-paracompact subset to 1 X furthermore it is fuzzy ω-paracompact subset to ̃ Proof: Let { ̃ } be any fuzzy covering of ̃ ∩ ̃ by fuzzy ω-open subset of 1 X since 1 X -̃ is fuzzy ω-open set in 1 X and ̃ -̃ ⊆ 1 X -̃ then for each ̃ -̃ there e ist fuzzy ω-open set ̃ in 1 X such that ̃ ⊆ ̃ -̃ and { ̃ } ⋃ { ̃} ̃ ̃ is a fuzzy covering of ̃ by fuzzy ω-open subset of 1 X , since ̃ is fuzzy ω-paracompact subset to 1 X , Therefore this cover has fuzzy ω-locally finite refinement { ̃ } , Which covers ̃ and ̃ is fuzzy ω-open set in 1 X for each that is the fuzzy ω-locally finite subfamily { ̃ } where ={ ; ̃ ⊆ ̃ for some } is fuzzy ω-open refinement of { ̃ } which covers ̃ ∩ ̃ too, thus ̃ ∩ ̃ is fuzzy ω-paracompact subset to 1 X , since ̃ is fuzzy ω-paracompact subset to 1 X so by 5.1 Proposition it is fuzzy ω-paracompact subspace of 1 X since ̃ fuzzy ω-closed in 1 X hence ̃ ∩ ̃ is fuzzy ω-closed subset of ̃ and then by 5.2 Proposition ̃ ∩ ̃ is fuzzy ω-paracompact subset to ̃