general topology – Does intersection of connected closed sets preserve connectedness?


Let $X$ be a compact, Hausdorff space, and $mathcal{A}={A_alpha}$ be a class of connected, closed subsets that forms a totally ordered set as in inclusion relation. I am trying to prove $tilde{A}=bigcap_{alpha}A_alpha$ is connected.

So here’s what I thought. Suppose $tilde{A}$ is not connected, which means $exists f:tilde{A}rightarrow {-1,1}$ that is a continuous surjection. Since $X$ is compact and Hausdorff therefore normal $(T_4)$, $f$ can be expanded to be a continuous $tilde{f}:Xrightarrow (-1,1) s.t.tilde{f}|_{tilde{A}}=f$.

From here I made two attempts:

First, from each $A_alpha$, I could pick a $x_alphain A_alpha$ forming a net. I wanted to prove ${f(x_alpha)}$ somehow converges to both $-1$ and $1$, which leads to contradiction, but it didn’t seem to work.

Second, by connectedness $tilde{f}(A_alpha)=(-1,1), forallalpha$, and if I could show $tilde{f}(bigcap A_alpha)=bigcap f(A_alpha)$ then the proof would be complete. Yet usually $f$ needs to be injective to satisfy the equation, so I was wondering if ${A_alpha}$ being a totally ordered set would be useful.

Any hint or help will be appreciated.