Union of Connected Sets with Non-Null Intersection is Connected

Theorem
Let $T = \left({X, \vartheta}\right)$ be a topological space.

Let $S = \bigcup A_\alpha \cup B$ where:
 * $\left\{{A_\alpha}\right\}$ is a set of subsets of $X$ where $\alpha$ is the element of some indexing set
 * $B \subseteq X$
 * All of $A_\alpha$ and $B$ are connected
 * $\forall \alpha: A_\alpha \cap B \ne \varnothing$

Then $S$ is connected.