Union of Open Irreducible Non-Disjoint Subspaces is Irreducible

Theorem
Let $T = \left({S, \tau}\right)$ be an irreducible toplogical space.

Let $U$ and $V$ be open irreducible subspaces.

Let their intersection $U \cap V$ be non-empty.

Then their union $U \cup V$ is irreducible.