Equivalence of Definitions of Path Component/Maximal Path-Connected Set is Union of Path-Connected Sets
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Theorem
Let $T = \struct {S, \tau}$ be a topological space.
Let $x \in T$.
Let $\tilde C$ be a maximal path-connected set of $T$ that contains $x$.
Then:
- $\tilde C = \bigcup \set {A \subseteq S : x \in A \land A \text{ is path-connected in } T}$.
Proof
Let $\CC_x = \set {A \subseteq S : x \in A \land A \text{ is path-connected in } T}$
Let $C = \bigcup \CC_x$
Lemma
- $C$ is path-connected in $T$ and $C \in \CC_x$.
By definition:
- $\tilde C \in \CC_x$
From Set is Subset of Union:
- $\tilde C \subseteq C$
By maximality of $\tilde C$:
- $\tilde C = C$
$\blacksquare$