\begin{definition}[Definition:Arc Component]
Let $T$ be a topological space.
Let us define the relation $\sim$ on $T$ as follows:
:$x \sim y \iff x$ and $y$ are arc-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 '''arc components of $T$'''.
If $x \in T$, then the '''arc component of $T$''' containing $x$ (that is, the set of points $y \in T$ with $x \sim y$) can be denoted by $\map {\operatorname {AC}_x} T$.
\end{definition}