Definition:Path Component/Equivalence Class

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

 * Path-Connectedness is Equivalence Relation


 * Equivalence of Definitions of Path Component