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

Then

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

Proof

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

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 $T$.

$\blacksquare$

Sources

• 2000: John M. Lee: Introduction to Topological Manifolds: $\S 4$ Connectedness and Compactness, Proposition $4.9 \ \text {(d)}$