Prove that the components of a closed set $E$ are closed.
For each point $a$, let $C(a)$ be the component containing $a$. Then $C(a)$ is the union of all connected sets containing $a$. We must prove that $C(a)$ is closed.
We have $a\in C(a)\subseteq E$. Since $C(a)$ is connected, we have that its closure $\overline{C(a)}$ is also connected. Also, $\overline{C(a)}\subseteq\overline{E}=E$ since $E$ is closed. So we have $a\in C(a)\subseteq \overline{C(a)}\subseteq\overline{E}$. We must prove that $C(a)=\overline{C(a)}$. How?
Best Answer
Answered by minar in the comments: since the Closure of a connected set is connected, $\overline{C(a)}$ is a connected set. By the definition of $C(a)$, $\overline{C(a)}\subseteq C(a)$, which means the equality holds.