# Definition:Arc (Topology)

## Definition

Let $T$ be a topological space.

Let $\mathbb I \subset \R$ be the closed unit interval $\closedint 0 1$.

Let $a, b \in T$.

An arc from $a$ to $b$ is a path $f: \mathbb I \to T$ such that $f$ is injective.

That is, an arc from $a$ to $b$ is a continuous injection $f: \mathbb I \to T$ such that $\map f 0 = a$ and $\map f 1 = b$.

The mapping $f$ can be described as an arc (in $T$) joining $a$ and $b$.

## Also known as

An arc is also known as an open curve.