[Math] Components are clopen in a space with a finite number of connected components

Question

I'm having trouble understanding why this fact is true. A lot of sites just assume it with out reason and it doesn't seem so direct to me.

Anyways, here is the theorem:

For any topological space $(X,T)$ with finite number of components, each component is clopen.

I only know the most basic definitions of components, basically that it is a maximal connected subset around a point.

Any help is appreciated.

Answer 1

A connected component $C$ of a space $X$ is always closed. This follows from the following fact:

Lemma: If $A$ is a connected subset of a topological space $X$, and $A\subseteq B\subseteq \overline A$, then $B$ is connected as well.

Now if $X$ has only finitely many components, then each component is complement of finitely many closed sets.