Fuzzy topological space

Definition
A fuzzy topological space is a set $$X$$ along with a collection $$\delta$$ of subsets of $$[0,1]^X$$ satisfying:


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

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