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

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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 $C = C'$.


Proof

\(\displaystyle y \in C'\) \(\leadstoandfrom\) \(\displaystyle \exists B \text{ a connected set of } T, x \in B, y \in B\) Definition of $\sim$
\(\displaystyle \) \(\leadstoandfrom\) \(\displaystyle \exists B \in \CC_x : y \in B\) Equivalent definition
\(\displaystyle \) \(\leadstoandfrom\) \(\displaystyle y \in \bigcup \CC_x\) Definition of Set Union
\(\displaystyle \) \(\leadstoandfrom\) \(\displaystyle y \in C\) Definition of $C$

The result follows.

$\blacksquare$


Also see