# Equivalence of Definitions of Component

## Contents

## Theorem

The following definitions of the concept of **Component** in the context of **Topology** are equivalent:

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

Let $x \in T$.

### Definition 1: Equivalence Class

From Connectedness of Points is Equivalence Relation, $\sim$ is an equivalence relation.

From the Fundamental Theorem on Equivalence Relations, the points in $T$ can be partitioned into equivalence classes.

These equivalence classes are called the **(connected) components** of $T$.

If $x \in S$, then the **component of $T$ containing $x$** (that is, the set of points $y \in S$ with $x \sim y$) is denoted by $\operatorname{Comp}_x \left({T}\right)$.

### Definition 2: Union of Connected Sets

The **component of $T$ containing $x$** is defined as:

- $\displaystyle \operatorname{Comp}_x \left({T}\right) = \bigcup \left\{{A \subseteq S: x \in A \land A}\right.$ is connected $\left.\right\}$

### Definition 3: Maximal Connected Set

The **component of $T$ containing $x$** is defined as:

- the maximal connected set of $T$ that contains $x$.

## Proof

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

Let $C = \bigcup \mathcal C_x$

### Lemma

- $C$ is connected in $T$ and $C \in \mathcal C_x$.

Let $C’$ be the equivalence class containing $x$ of the equivalence relation $\sim$ defined by:

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

### Equivalence Class equals Union of Connected Sets

It needs to be shown that $C = C’$.

\(\displaystyle y \in C’\) | \(\leadstoandfrom\) | \(\displaystyle \exists B \text{ a connected set of } T, x \in B, y \in B\) | Definition of $\sim$ | ||||||||||

\(\displaystyle \) | \(\leadstoandfrom\) | \(\displaystyle \exists B \in \mathcal C_x : y \in B\) | Equivalent definition | ||||||||||

\(\displaystyle \) | \(\leadstoandfrom\) | \(\displaystyle y \in \bigcup \mathcal C_x\) | Definition of Set Union | ||||||||||

\(\displaystyle \) | \(\leadstoandfrom\) | \(\displaystyle y \in C\) | Definition of $C$ |

The result follows.

$\Box$

### Union of Connected Sets is Maximal Connected Set

Let $\tilde C$ be any connected set such that:

- $C \subseteq \tilde C$

Then $x \in \tilde C$.

Hence $\tilde C \in \mathcal C_x$.

From Set is Subset of Union,

- $\tilde C \subseteq C$.

Hence $\tilde C = C$.

It follows that $C$ is a maximal connected set of $T$ by definition.

$\Box$

### Maximal Connected Set is Union of Connected Sets

Let $\tilde C$ be a maximal connected set of $T$ that contains $x$.

By definition, $\tilde C \in \mathcal C_x$.

From Set is Subset of Union, $\tilde C \subseteq C$.

By maximality of $\tilde C$ then $\tilde C = C$

$\blacksquare$

## Also see

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
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