Union of Connected Sets with Non-Empty Intersections is Connected/Corollary

Corollary to Space with Connected Intersection has Connected Union
Let $T = \left({S, \tau}\right)$ be a topological space.

Let $I$ be an indexing set.

Let $\mathcal A = \left \langle{A_\alpha}\right \rangle_{\alpha \mathop \in I}$ be an indexed family of subsets of $S$, all connected in $T$.

Let $B$ be a connected set of $T$ such that:
 * $\forall C \in \mathcal A: B \cap C \ne \varnothing$

Then $\displaystyle B \cup \bigcup \mathcal A$ is connected.