# Definition:Path Component

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## Contents

## Definition

Let $T$ be a topological space.

### 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 $\operatorname{PC}_x \left({T}\right)$.

### 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 $x$.