# Definition:Path Component

## 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 $x$.