Equivalence of Definitions of Component/Maximal Connected Set is Union of Connected Sets

Theorem

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

Let $x \in T$.

Let $\tilde C$ be a maximal connected set of $T$ that contains $x$.

Then:

$\tilde C = \bigcup \set {A \subseteq S : x \in A \land A \text{ is connected in } T}$.

Proof

Let $\CC_x = \set {A \subseteq S : x \in A \land A \text{ is connected in } T}$

Let $C = \bigcup \CC_x$

Lemma

$C$ is connected in $T$ and $C \in \CC_x$.

$\Box$

By definition:

$\tilde C \in \CC_x$

From Set is Subset of Union:

$\tilde C \subseteq C$

By maximality of $\tilde C$:

$\tilde C = C$

$\blacksquare$

Also see

• Union of Connected Sets is Maximal Connected Set