# 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.

$\blacksquare$