show existence of a finite set $\{K_{j_1},....K_{j_m}\}$ with empty intersection

Question

Let $\{K_\alpha\}$ be a family of compact sets in the Hausdorff topological space $X$, such that $\cap_{\alpha\in I}{K_\alpha}=\emptyset$...

Is there necessarily a finite set $\{K_{j_1},....K_{j_m}\}$ such that $\cap_{n=1}^m{K_{j_n}}=\emptyset$? if so... why? I can see why it is true on metric spaces, but cannot conclude it for general topology....

Answer 1

Pick $j_0\in I$. Then the open sets $X-K_\alpha$ cover $K_{j_0}$. Hence there is a finite sub-cover: $$K_{j_0}\subseteq \bigcup _{n=1}^m(X-K_{j_n}).$$ But then $$ K_{j_0}\cap \bigcap _{n=1}^mK_{j_m}=\emptyset.$$