# 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.