Definition:Path Component

Definition
Let $T$ be a topological space.

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


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

We have that $\sim $ is an equivalence relation, so 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 $\operatorname{PC}_x \left({T}\right)$.