Properties of compact set: non-empty intersection of any system of closed subsets with finite intersection property Let $X$ be a Hausdorff topological vector space. Let $C$ be a nonempty compact subset of $X$ and $\{C_\alpha\}_{\alpha \in I}$ be a collection of closed subsets such that $C_\alpha \subset C$ for each $\alpha \in I$ with $I$ is an infinite index set. Assume that the intersection of any finite sets among $\{C_\alpha\}_{\alpha \in I}$ is nonempty. Can we conclude that $\bigcap_{\alpha \in I}C_\alpha \neq \emptyset$?
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The more general fact is true. If $(C_\alpha)_{\alpha\in I}$ is a collection of closed subsets with finite intersection property, then all these subsets have a common point.
Assume $\cap_{\alpha\in I}C_\alpha=\varnothing$, then for open subsets of $C$ which we denote $U_\alpha=C\setminus C_\alpha$ we have $\cup_{\alpha\in I}U_\alpha=C$. Since $C$ is compact, we have finite collection $\{\alpha_1,\ldots,\alpha_n\}\subset I$ such that $C=U_{\alpha_1}\cup\ldots\cup U_{\alpha_n}$. Taking complements we get that $C_{\alpha_1}\cap\ldots C_{\alpha_n}=\varnothing$. Contradiction, so $\cap_{\alpha\in I}C_\alpha\neq\varnothing$
