Union of Connected Sets with Common Point is Connected/Proof 1

Proof
Let $B = \ds \bigcup_{\alpha \mathop \in A} B_\alpha$.

From definition 3 of a connected set, $B$ is connected in $T$ the subspace $\struct {B, \tau_B}$ is a connected space.

From definition 4 of a connected space, $\struct {B, \tau_B}$ is connected the only clopen sets in $\struct {B, \tau_B}$ are $B$ and $\O$.

Let $U$ be any clopen set of the subspace of $\struct {B, \tau_B}$.

Let $V = B \setminus U$.

From Complement of Clopen Set is Clopen, $V$ is also clopen.

Hence $U, V$ are disjoint clopen sets such that $B = U \cup V$.

assume $x \in U$.

From Set is Subset of Union:
 * $\ds \forall \alpha \in A : B_\alpha \subseteq B = U \cup V$

From Connected Subset of Union of Disjoint Open Sets:
 * $\forall \alpha \in A : B_\alpha \subseteq U$

From Union is Smallest Superset:
 * $B \subseteq U$

Since $U \subseteq B$ then $U = B$.

From Set Difference with Self is Empty Set,
 * $V = B \setminus U = B \setminus B = \O$

Hence the only clopen sets in $\struct {B, \tau_B}$ are $B$ and $\O$.

The result follows.