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 $\delta$ 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.