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