A note on weak fuzzy P-spaces

In this paper, the concepts of weak fuzzy P -spaces, are extensively established. The inter-relations of weak fuzzy P -spaces, fuzzy P -spaces, fuzzy almost P -spaces, fuzzy weakly Lindelöf spaces and fuzzy almost Lindelöf spaces are also investigated in this paper.


Introduction
The concept of fuzzy sets and fuzzy set operations were first introduced by L.A.Zadeh in his classical paper [16] in the year 1965. Thereafter the paper of C.L.Chang [5] in1968 paved the way for the subsequent tremendous growth of the numerous fuzzy topological concepts. Since then much attention has been paid to generalize the basic concepts of General Topology in fuzzy setting andthus a modern theory of fuzzy topology has been developed. In recent years, fuzzy topology has been found to be very useful in solving many practical problems. In this paper, the concepts of weak fuzzy P -spaces, are extensively established. The inter-relations of weak fuzzy P -spaces, fuzzy P -spaces, fuzzy almost P -spaces, fuzzy weakly Lindelöf spaces and fuzzy almost Lindelöf spaces are also investigated in this paper. Let X be a non-empty set and I, the unit interval [0,1]. A fuzzy set λ in X is a function from X into I. The null set 0 is the function from X into I which assumes only the value 0 and the whole fuzzy set 1 is the function from X into I which takes 1 only.
Definition 2.1 [5] Let λ and µ be fuzzy sets in X. Then for all x ∈ X, For a family {λ i /i ∈ I} of fuzzy sets in X, the union ψ = ∨ i λ i and intersection The fuzzy set 0 X is defined as 0 X (x) = 0, for all x ∈ X and the fuzzy set 1 X defined as 1 X (x) = 1, for all x ∈ X.
Definition 2.2 [6] A fuzzy point x α in X is a fuzzy set with membership function defined as:

Definition 2.3 [5]
A fuzzy topology is a family 'T 'of fuzzy sets in X which satisfies the following conditions: T is called a fuzzy topology for X and the pair (X, T ) is a fuzzy topological space or fts in short. Definition 2.4 [5] Let (X, T ) be any fuzzy topological space and λ be any fuzzy set in (X, T ). The closure and interior of a fuzzy set λ in a fuzzy topological space (X, T ) are respectively denoted as cl(λ) and int(λ) are defined as Lemma 2.5 [15] For a fuzzy set λ of a fuzzy space X, For a family A = {λ a } of fuzzy sets of a fuzzy space X, ∨ cl λ α ≤ cl(∨λ α ). In case A is a finite set, ∨cl(λ α ) = cl(∨λ α ). Also ∨ int λ α ≤ int(∨λ α ).
Definition 2.9 [13] A fuzzy set λ in a fuzzy topological space (X, T ) is called a fuzzy dense set if there exists no fuzzy closed set µ in (X, T ) such that λ < µ < 1.

Definition 2.10 [8]
A non-zero fuzzy set λ in a fuzzy topological space (X, T ) is called a fuzzy somewhere dense set if intcl(λ) = 0 in (X, T ).
Definition 2.11 [14] Let (X, T ) be a fuzzy topological space. A fuzzy set λ in (X, T ) is called a fuzzy σ-nowhere dense set if λ is a fuzzy F σ -set in (X, T ) such that int(λ) = 0. Definition 2.13 [12] A fuzzy topological space (X, T ) is called a fuzzy almost Lindelöf space if every fuzzy open cover {λ α } α∈∆ of (X, T ) admits a countable subcover {λ n } n∈N such that ∨ n∈N cl(λ n ) = 1.
s are fuzzy regular closed sets in (X, T ), is fuzzy regular closed in (X, T ).
Theorem 2.20 [9] If a fuzzy topological space (X, T ) is a weak fuzzy P -space, then 4 Characterizations of weak fuzzy P -spaces s are fuzzy open sets in a weak fuzzy P -space (X, T ) and λ = 1, then λ is not a fuzzy dense set in (X, T ).
s are fuzzy open sets in a weak fuzzy P -space (X, T ) and λ = 1, then 1 − λ is a fuzzy somewhere dense set in (X, T ).
s are fuzzy open sets in a weak fuzzy P -space (X, T ) and λ = 1, then λ is not a fuzzy σ-nowhere dense set in (X, T ).
s are fuzzy open sets in (X, T ), implies that (cl(λ i ))'s are fuzzy regular closed sets in (X, T ). Also, since fuzzy regular closed sets are fuzzy closed sets in a fuzzy topological space, is a fuzzy regular closed set in (X, T ) and hence λ is a fuzzy regular closed set in (X, T ). Then, clint(λ) = λ, in (X, T ). This implies that int(λ) = 0, in (X, T ). Thus, λ is a fuzzy F σ -set in (X, T ) such that int(λ) = 0 in (X, T ), implies that λ is not a fuzzy σ-nowhere dense set in (X, T ).
Remark 5.6 (1) Clearly every fuzzy P -space is a fuzzy almost P -space, since for every non-zero fuzzy G δ -set δ in (X, T ), int(δ) = δ = 0. But the converse need not be true. For, in example (5.5), for every non-zero fuzzy G δ -set δ in (X, T ), int(δ) = 0 in (X, T ). Hence (X, T ) is a fuzzy almost P -space, but (X, T ) is not a fuzzy P -space, since the fuzzy 1} is a fuzzy topology on X. In (X, T ), for every non-zero fuzzy G δ -set δ, int(δ) = 0 in (X, T ). Hence (X, T ) is a fuzzy almost P -space, but (X, T ) is not a weak fuzzy P -space.

Conclusions
In this paper, the concepts of weak fuzzy P -spaces, were extensively established and also inter-relations between weak fuzzy P -spaces and other fuzzy topological spaces, were investigated.