Union of Connected Sets with Non-Empty Intersections is Connected/Corollary/Proof 1

Corollary to 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 $B$ be a connected set of $T$ such that:

$\forall C \in \AA: B \cap C \ne \O$

Then $\ds B \cup \bigcup \AA$ is connected.

Proof

Let $C \in \AA$.

From Union of Connected Sets with Non-Empty Intersections is Connected applied to $B$ and $C$, the union $B \cup C$ is connected.

Thus the set $\tilde \AA = \set {B \cup C: C \in \AA}$ satisfies the conditions of the theorem.

Hence the result.

$\blacksquare$

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $6.2$: Connectedness: Corollary $6.2.16$