Extensions Of Uniform Space Notions

A large part of mathematics is based on the notion of a set and on binary logic. Statements are either true or false and an element either belongs to a set or not. In order to accommodate the idea of a sliding transition between the two states: true and false, and to generalise the concept of a subset of a given set, Zadeh introduced the notion of a fuzzy subset in a now-famous paper: [52]. For the record, let us recall that if X is a set and A is a subset of X then the characteristic function, denoted 1 A , is defined by Thus 1 A ∈ 2 X . In [52], an element μ ∈ I X , where I denotes the closed unit interval, was called a fuzzy set in X, with μ(x) being interpreted as the degree to which x belongs to the fuzzy set μ. Since the elements μ ∈ I X are generalisations of subsets of X, it is more accurate to refer to them as fuzzy subsets of X and we shall adopt this terminology here.

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