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

Corollary to Spaces with Connected Intersection have Connected Union
Let $T$ be a topological space.

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

Suppose there is a connected subspace of $T$ such that $B \cap C \ne \varnothing$ for all $C \in \mathcal A$.

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

Proof
Let $C \in \mathcal A$.

From Spaces with Connected Intersection have Connected Union applied to $B$ and $C$, the union $B \cup C$ is connected.

Thus the set $\tilde{\mathcal A} = \left\{{B \cup C: C \in \mathcal A}\right\}$ satisfies the conditions of the theorem, and the claim follows.