Union of Connected Sets with Common Point is Connected

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

Let $\set {B_\alpha}_{\alpha \in A}$ be a family of connected subspaces of $T$.

Let $x \in \bigcap \set{B_\alpha: \alpha \in A}$.

Then
 * $\bigcup \set{B_\alpha: \alpha \in A}$ is a connected subspace of $T$.