Theorem 7.5

Theorem 7.5 (Finite intersection characterization). A space $X$ is compact if and only if every family of closed sets $\{C_i:i\in I\}$ with the finite intersection property has nonempty total intersection. The finite intersection property means $$C_{i_1}\cap\cdots\cap C_{i_k}\ne\varnothing$$ for every finite subfamily.