Union of Simply Connected Sets with Path-Connected Intersection is Simply Connected

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

Consider all subsets of $T$ as subspaces, equipped with the subspace topology induced by $\tau$.

Let $U$ and $V$ be open subsets of $T$ that are simply connected.

Let $U \cap V$ be non-empty and path-connected.

Let $U \cup V = T$.

Then $\struct {T, \tau}$ is simply connected.