Irreducible Subspace is Contained in Irreducible Component

Theorem
Let $T = \struct {S_1, \tau}$ be a topological space.

Let $H = \struct {S_2, \tau_H}$ be an irreducible subspace of $T$.

Then there exists an irreducible component $C = \struct {S_3, \tau_C}$ of $T$ such that $S_2 \subseteq S_3$.

Outline of Proof
We apply Zorn's Lemma to the set of irreducible subspaces, ordered by the subset relation.

Proof
By definition, an irreducible component of $X$ is an irreducible subspace that is maximal among the irreducible subspaces, ordered by the subset relation.

Also see

 * Point is Contained in Irreducible Component