# Equivalence of Definitions of Path Component

## Contents

## Theorem

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

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

Let $x \in T$.

### Equivalence Class

Let $\sim$ be the equivalence relation on $T$ defined as:

- $x \sim y \iff x$ and $y$ are path-connected.

The equivalence classes of $\sim$ are called the **path components of $T$**.

If $x \in T$, then the **path component of $T$** containing $x$ (that is, the set of points $y \in T$ with $x \sim y$) can be denoted by $\map {\operatorname{PC}_x} T$.

### Union of Path-Connected Sets

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

- $\displaystyle \map {\operatorname{PC}_x} T = \bigcup \left\{{A \subseteq S: x \in A \land A}\right.$ is path-connected $\left.\right\}$

### Maximal Path-Connected Set

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

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

## Proof

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

Let $C = \bigcup \CC_x$.

### Lemma

- $C$ is path-connected in $T$ and $C \in \CC_x$.

$\Box$

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 Path-Connected Sets

It needs to be shown that $C = C'$.

\(\displaystyle y \in C'\) | \(\leadstoandfrom\) | \(\displaystyle x \text{ is path-connected to } y \text{ in } T\) | Definition of $\sim$ | ||||||||||

\(\displaystyle \) | \(\leadstoandfrom\) | \(\displaystyle \exists B \text{ a connected set of } T, x \in B, y \in B\) | Points are Path-Connected iff Contained in Path-Connected Set | ||||||||||

\(\displaystyle \) | \(\leadstoandfrom\) | \(\displaystyle \exists B \in \CC_x : y \in B\) | 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.

$\Box$

### Union of Path-Connected Sets is Maximal Path-Connected Set

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

- $C \subseteq \tilde C$

Then $x \in \tilde C$.

Hence $\tilde C \in \CC_x$.

From Set is Subset of Union,

- $\tilde C \subseteq C$.

Hence $\tilde C = C$.

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

$\Box$

### Maximal Path-Connected Set is Union of Path-Connected Sets

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

By definition:

- $\tilde C \in \CC_x$

From Set is Subset of Union:

- $\tilde C \subseteq C$

By maximality of $\tilde C$:

- $\tilde C = C$

$\blacksquare$

## Also see

## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $4$: Connectedness