Irreducible Components of Hausdorff Space are Points

Theorem

Let $T = \struct {S, \tau}$ be a non-empty Hausdorff space.

Then the irreducible components of $T$ are the singleton sets.

Proof

By Subspace of Hausdorff Space is Hausdorff, the irreducible components of $T$ are also Hausdorff.

By Irreducible Hausdorff Space is Singleton, they can only be singletons.

By Trivial Topological Space is Irreducible, every singleton of $X$ is indeed irreducible.

$\blacksquare$