Definition:Arc Component

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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 $\map {\operatorname {AC}_x} T$.