Definition:Path Component/Equivalence Class

Definition
Let us define the relation $\sim$ on $T$ as follows:


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

That is, there exists a continuous mapping $f: \closedint 0 1 \to S$ such that $\map f 0 = x$ and $\map f 1 = y$. 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.

These equivalence classes 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 $\operatorname{PC}_x \left({T}\right)$.