Definition:Arc (Topology)

Definition

Let $T$ be a topological space.

Let $I \subset \R$ be the unit interval $\left[{0 . . 1}\right]$.

Let $a, b \in T$.

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

That is, an arc from $a$ to $b$ is a continuous injection $f: I \to T$ such that $f \left({0}\right) = a$ and $f \left({1}\right) = b$.

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

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 4$