Union of Path-Connected Sets with Common Point is Path-Connected

Theorem
Let $T = \struct {S, \tau}$ be a topological space.

Let $\family {B_\alpha}_{\alpha \mathop \in A}$ be a family of path-connected sets of $T$.

Let $\exists x \in \displaystyle \bigcap \family {B_\alpha}_{\alpha \mathop \in A}$.

Then
 * $\displaystyle \bigcup \family {B_\alpha}_{\alpha \mathop \in A}$ is a path-connected set of $T$.

Proof
Let $B = \displaystyle \bigcup \family {B_\alpha}_{\alpha \mathop \in A}$.

Let $a, b \in B$.

Then
 * $\exists \alpha, \beta \in A: a \in B_\alpha \land b \in B_\beta$.

As $B_\alpha$ is a path-connected set in $T$ then $a$ and $x$ are path-connected points.

Similarly, $x$ and $b$ are path-connected points.

From Joining Paths makes Another Path, $a$ and $b$ are path-connected points.

Since $a$ and $b$ were arbitrary points then $B$ is a path-connected set of $T$.