Union of Connected Sets with Non-Empty Intersections is Connected/Corollary/Proof 3

Proof
Let $\ds H = B \cup \bigcup \AA$

that $H$ is not connected in $T$.

That is, that $H$ is disconnected.

From the definition of disconnected, there exist separated sets $U, V$ whose union is $H$.

It follows that $B$ is disconnected.

This contradicts our condition that $B$ is a connected set in $T$.

It follows by Proof by Contradiction that $H$ is connected in $T$.