Definition:Arc (Topology)

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Definition

Let $T$ be a topological space.

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

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 $f \left({0}\right) = a$ and $f \left({1}\right) = b$.


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


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