Union_of_Connected_Sets_with_Non_Empty_Intersections_is_Connected

Let $T = \struct {S, \tau}$ be a topological space. Let $I$ be an indexing set. Let $\AA = \family {A_\alpha}_{\alpha \mathop \in I}$ be an indexed family of subsets of $S$, all connected in $T$. Let $\AA$ be such that no two of its elements are disjoint $\forall B, C \in \AA: B \cap C \ne \O$ Then $\displaystyle \bigcup \AA$ is itself connected.