If U contains some point in $A_{\alpha'}$, and $A_{\alpha'}$ is connected, then $A_{\alpha'} \subset U$.

Question

This is a theorem 5.5, in Croom, topology textbook.

I have a problem with red line. How this can be justified?

If U contains some point in $A_{\alpha'}$, and $A_{\alpha'}$ is connected, then $A_{\alpha'} \subset U$.

I have trouble with understanding above statement.

Answer 1

We know $A_{\alpha'}\subseteq U\cup V$. If $A_{\alpha'}\not\subseteq U$, then $A_{\alpha'}\cap V$ would be nonempty, and $A_{\alpha'}\cap U,A_{\alpha'}\cap V$ would be a partition of $A_{\alpha'}$ into nonempty open sets, contradicting $A_{\alpha'}$ being connected.