Irreducible Subspace is Contained in Irreducible Component

Theorem
Let $X$ be a topological space.

Let $Y\subset X$ be an irreducible subspace.

Then there exists an irreducible component of $X$ such that $Y \subseteq X$.

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