Fuzzy pre-continuous and fuzzy pre*-continuous function of fuzzy pre-compact space in fuzzy topological space

The aim of this paper is to introduce and study the notion of a fuzzy pre-continuous function, fuzzy pre*continuous function, fuzzy pre-compact space and some properties, remarks related to them.


Introduction
The concept of fuzzy set was introduced by Zadeh (1965) in classical paper. Chang (1968)

Fuzzy pre-compact space
In this section we study some definitions, remarks, Propositions and theorems about fuzzy Pre-compact space in fuzzy topological spaces. Definition 2.1: [1] [2] A collection T ̃ of a sub set of A ̃ that is T ̃ ⊆ p (A ̃) is said to be fuzzy topology on A ̃i f satisfies the following condition: If every fuzzy pre-open subset of a fuzzy topological space (Ã , T) is a fuzzy pre-compact, then every subset of Ã is a fuzzy pre-compact. Proof: Obvious ■ Proposition (2.10) [4] A fuzzy pre-closed subset of fuzzy pre-compact space is fuzzy pre-compact Proof: Suppose that (Ã , T) be a fuzzy pre-compact space And B be a fuzzy pre-closed subset of (Ã , T) Hence B is fuzzy pre-compact space ■ Corollary (2.11) [4] A fuzzy closed subset of a fuzzy pre-compact is fuzzy compact. Proof: Thus, Ã ⋃ B is fuzzy pre compact ■ Remark 2.14: [8] If B and C are a fuzzy pre-compact subsets of a fuzzy topological space (Ã ,T) then B ⋂ C is need not to be fuzzy pre-compact space.

Proof:
Suppose that ̃ is fuzzy pre-compact subspace in ̃ ∴ ̃ is fuzzy pre-compact relative to ̃ And ̃ is fuzzy pre-compact relative to ̃ Hence ̃ is fuzzy pre-compact in ̃ Conversely Let ̃ is fuzzy pre-compact in ̃ ∴ ̃ is fuzzy pre-compact relative to ̃ Then ̃ is fuzzy pre-compact relative to ̃ Hence ̃ is fuzzy pre-compact in ̃ ■

Fuzzy pre-continuous and fuzzy pre*-continuous
Since f is fuzzy continuous Then f is fuzzy pre* -continuous ■ Since Let f: ( Ã , T ) → (̃ ,̃`) be a fuzzy pre-continuous surjective function and Ã is a fuzzy pre-closed compact then B is a fuzzy pre-closed compact. Proof: Obvious ■ 3) Let if f: ( Ã , T ) → (̃ ,̃`) be a fuzzy -pre continues surjective function of a fuzzy pre-compact a space Ã onto a space B then B is fuzzy pre-compact.
Proof: Obvious ■ 4) Let if f: ( Ã , T ) → (̃ ,̃`) be a fuzzy -pre continues bijective function and B be a fuzzy pre-compact space Ã then B is fuzzy Pre-compact. Proof Obvious ■ Remark 3.13: [7] The following diagram explains the relationships among the different types of fuzzy continuous function.

Conclusion
It is an interesting exercise to work on fuzzy, pre-continuous and fuzzy pre*-continuous; function of fuzzy pre-compact space in fuzzy topological space similarly other forms of fuzzy pre-open set can be applied to difine different forms of fuzzy pre-compact space.