prove that intersection of the family of the set is connected

Question

I got no clue to solve this problem, because I can't find the connection between the compactness and the connectedness for the set family. Can anyone help me to solve this? I really appreciate.

$X$ is Hausdorff space and $(A_\alpha)_{\alpha \in J}$ is the family of compact subsets of $X$. Prove that if for all finite subset $I$ of $J$ we have $\bigcap_{\alpha \in I}{A_\alpha}$ connected, then $\bigcap_{\alpha \in J}{A_\alpha}$ connected.

Answer 1

Let $K=\bigcap_{\alpha\in J}A_\alpha$; $K$ is compact, and $X$ is Hausdorff, so $K$ is closed. If $K$ is not connected, it can be partitioned into non-empty closed sets $H$ and $K$. $H$ and $K$ are disjoint compact sets in a Hausdorff space, so there are disjoint open sets $U$ and $V$ in $X$ such that $H\subseteq U$ and $K\subseteq V$. (If you don’t know this result, prove it first in the special case that one of $H$ and $K$ is a singleton, then use that result to prove the general one.)

Now $U\cup V$ is an open nbhd of $K$. Prove that there is a finite $I\subseteq J$ such that $\bigcap_{\alpha\in I}A_\alpha\subseteq U\cup V$, and conclude that $\bigcap_{\alpha\in I}A_\alpha$ is not connected. If you get stuck on this part of the proof, check the spoiler-protected hint below.

Let $W=U\cup V$, and for $\alpha\in J$ let $W_\alpha=X\setminus A_\alpha$. Fix $\beta\in J$. If $A_\beta\subseteq W$, we’re done. If not, and let $\mathscr{W}=\{W\}\cup\{W_\alpha:\alpha\in J\setminus\{\beta\}\}$; $\mathscr{W}$ is an open cover of the compact set $A_\beta$.