Fuzzy topological space

This is a variation of topological space. View other variations of topological space

Definition

A fuzzy topological space is a set $X$ along with a collection $δ$ of subsets of $[0, 1]^{X}$ satisfying:

The constant functions $0$ and $1$ are in $δ$ .
For any two functions $A, B \in δ$ , the function $A \land B$ , which sends each $x \in X$ to $min {A (x), B (x)}$ , is also in $δ$ .
For any collection $A_{i}, i \in I$ , of members of $δ$ , the function $\lor_{i} A_{i}$ , which sends $x \in X$ to $max A_{i} (x)$ , is also in $δ$ .

Such a collection of subsets is termed a fuzzy topology. The members of $δ$ are fuzzy open sets.