On fuzzy function spaces

Gunther Jäger

doi:10.1155/s0161171299227275

Abstract

In [3], we started the investigation of compactness in fuzzy function spaces in FCS, the category of fuzzy convergence spaces as defined by Lowen/Lowen/Wuyts [8]. This paper goes somewhat deeper in the investigation of fuzzy function spaces using the notion of splitting and conjoining structures on fuzzy subsets. We discuss the connection to the exponential law and give several examples of such structures. As a special case, we study a notion of fuzzy compact open topology.

Highlights

The theory of fuzzy topological spaces is highly developed
This paper takes FCS as a starting point to discuss certain fuzzy function space structures via splitting and conjoining structures. It continues a previous paper by the author, where compactness in fuzzy function spaces in FCS was considered [3] and considers function spaces in FTS
For two fuzzy convergences lim, lim on the same fuzzy set A ∈ [0, 1]X, we say that lim is finer than lim if and only if, for every prime fuzzy filter F ∈ F(A), we have lim F ⊂ lim F

Summary

Introduction

The theory of fuzzy topological spaces is highly developed. Especially, compactness and separation axioms have been thoroughly studied. For two fuzzy convergences lim, lim on the same fuzzy set A ∈ [0, 1]X , we say that lim is finer than lim if and only if, for every prime fuzzy filter F ∈ F(A), we have lim F ⊂ lim F. Continuous convergence on fuzzy subsets (The space C(A, B)) Let in this number A ∈ [0, 1]X , B ∈ [0, 1]Y , and always sup A(X) 1B0 ⊂ B. For (A, limA), (B, limB) fcs’s, g : A → B a fuzzy mapping and F ∈ F(BA) a prime fuzzy filter, we put. 4.2], we showed that, for a continuous fuzzy mapping g : (A, limA) → (B, limB) and for 0 < α ≤ η(g), we have α1g ⊂ c- lim[α1g]. The reverse inclusion follows using the continuity of the evaluation map (Proposition 3.2) and the continuity of the fuzzy product-mapping (Proposition 2.2) in exactly the same way as the proof of the corresponding “classical” theorem 2.2, (a)⇒(b) in Poppe [12]

Splitting and conjoining fuzzy convergences

Examples for splitting and conjoining fuzzy convergences