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

Theorem

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

Let $x \in T$.

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

Let $C = \bigcup \CC_x$

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

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

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

Then $C = C'$.

$\ds y \in C'$

$\leadstoandfrom$

$\ds \exists B \text{ a connected set of } T, x \in B, y \in B$

Definition of $\sim$

$\ds $

$\leadstoandfrom$

$\ds \exists B \in \CC_x : y \in B$

equivalent definition

$\ds $

$\leadstoandfrom$

$\ds y \in \bigcup \CC_x$

$\ds $

$\leadstoandfrom$

$\ds y \in C$

Definition of $C$

The result follows.

$\blacksquare$