\begin{definition}[Definition:Path Component/Equivalence Class]
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.
\end{definition}