Equivalence of Definitions of Path Component/Equivalence Class equals Union of Path-Connected Sets

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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 $C = \bigcup \CC_x$


Let $\sim$ be the equivalence relation defined by:

$y \sim z$ if and only if $y$ and $z$ are path-connected in $T$.

Let $C'$ be the equivalence class of $\sim$ containing $x$.


Then $C = C'$.


Proof

\(\ds y \in C'\) \(\leadstoandfrom\) \(\ds x \text{ is path-connected to } y \text{ in } T\) Definition of $\sim$
\(\ds \) \(\leadstoandfrom\) \(\ds \exists B \text{ a connected set of } T, x \in B, y \in B\) Points are Path-Connected iff Contained in Path-Connected Set
\(\ds \) \(\leadstoandfrom\) \(\ds \exists B \in \CC_x : y \in B\) Equivalent definition
\(\ds \) \(\leadstoandfrom\) \(\ds y \in \bigcup \CC_x\) Definition of Union of Set of Sets
\(\ds \) \(\leadstoandfrom\) \(\ds y \in C\) Definition of $C$

The result follows.

$\blacksquare$


Also see