User:Zero sharp/Finite intersection property

Definition. Let $ X$ be set, and let $ A=\{A_i\}_{i\in I}$ be a collection of subsets in $ X$. Then $ A$ has the finite intersection property, if for any finite $ J\subset I$, the intersection $ \bigcap_{i\in J} A_i$ is non-empty.

A topological space $ X$ has the finite intersection property if the following implication holds: If $ \{A_i\}_{i\in I}$ is a collection of closed subsets in $ X$ with the finite intersection property, then the intersection $ \bigcap_{i\in I}A_i$ is non-empty.