Connected Components are Closed

Solution 1:

A component of $x$ is the largest connected set containing $x$.

Let $C$ be a component of $x$. Thus $x\in\overline{C}$ and $C \subseteq \overline{C}$.

However, $C$ is the largest connected set, therefore $\overline{C} \subseteq C$.
Hence $C = \overline{C}$, and $C$ is closed.

Solution 2:

Your proof is about right. I'd formulate it as follows: a connected component $C$ of $X$ is a maximally connected subset; this means 2 things:

1. $C$ is connected.
2. if $C \subseteq D$ and $D$ is connected, $C=D$.

Now use that $C$ connected implies $\overline{C}$ connected. Then applying 2. and noting that obviously $C \subseteq \overline{C}$, we conclude that $C = \overline{C}$, which is equivalent to $C$ being closed. QED.