Connected components are closed

Statement

Let $X$ be a topological space and let $x \in X$. The connected component of $x$ is the union $C(x)$ of all connected subsets of $X$ that contain $x$. Then $C(x)$ is connected and closed in $X$.

Remarks

Proof: a union of connected sets with a common point is connected, so $C(x)$ is the largest connected set that contains $x$. The closure of $C(x)$ is also connected, so it equals $C(x)$. The components of $X$ form a partition of $X$. Components need not be open. In $\mathbb{Q}$, the components are the single points, which are not open. If $X$ has only finitely many components, then each of them is also open.

Explanations

L1 human

The component of a point is the largest piece of the space that contains the point and is in one piece. Taking the closure cannot make the piece larger, so each component is closed.