I've read that for $A$ a connected subset of a metric space $M$ and $C$ a clopen (closed and open) subset of $M$, one could prove that either $A \subset C$ or $A \cap C=\varnothing$ and use it to prove that $\overline{A}$ is a connected subset of $M$.
I don't see how neither of the proofs can be done, I'm guessing I haven't spotted the right idea...
Can you help me please? I'd like to figure this out!
Thank you very much!
For the first one, suppose not, then $A\cap C\neq \emptyset$ and $A\setminus C\neq \emptyset$, which are open set in $A$ separates $A$. Then it contradicts $A$ is connected.
For the second one, you can prove if $A$ is connected, $x$ is a limit point of $A$, then $A\cup \{x\}$ is also connected, since if it's not, there are two open sets $G_1,G_2$ separate it. Then $A$ must be inside one of them, say $G_1$ and $x\in G_2$, then it will contradict the fact $x$ be a limit point. And $\bar{A}$ is just $A$ unions all its limit points.