# Definition:Path Component/Equivalence Class

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## Definition

Let $T$ be a topological space.

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 $\map {\operatorname{PC}_x} T$.

From Path-Connectedness 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.

## 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

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $6.5$: Components: Definition $6.5.5$