Union of Connected Sets with Non-Empty Intersections is Connected

Theorem
Let $T$ be a topological space.

Let $\mathcal A$ be a set of connected subspaces of $T$.

Suppose that, for all $B, C \in \mathcal A$, the intersection $B \cap C$ is non-empty.

Then $A = \bigcup \mathcal A$ is itself connected.