Union of Connected Sets with Common Point is Connected

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

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

Let $x \in \displaystyle \bigcap \family {B_\alpha}_{\alpha \mathop \in A}$.

Then
 * $\displaystyle \bigcup \family {B_\alpha}_{\alpha \mathop \in A}$ is a connected set of $T$.