Modifiers connect L-fuzzy sets to topological spaces

Abstract

In this paper we connect lattice-valued fuzzy sets, L-fuzzy sets, to the so-called accessibility relations which are at least reflexive relations on a non-empty set X. Because weakening and substantiating modifiers are defined by accessibility relations, we define them by L-fuzzy sets in this context. This type of modifiers induce topological spaces (see [7]) and we can say that L-fuzzy sets now induce topological spaces. We will show that all the level sets of an L-fuzzy set are open sets in the induced topology. We will also show that for any topology on a non-empty finite set X there exists a unique accessibility relation. We will also define the connectedness of a fuzzy set and show that normal sets are special types of connected sets. These ideas will lead us to define new relations between L-fuzzy sets, namely, weak equivalence, topological similarity and fineness relation.