# Definition:Arc-Connected/Subset

< Definition:Arc-Connected(Redirected from Definition:Arc-Connected Set)

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## Definition

Let $T = \left({S, \tau}\right)$ be a topological space.

Let $U \subseteq S$ be a subset of $T$.

Let $T\,' = \left({U, \tau_U}\right)$ be the subspace of $T$ induced by $U$.

Then $U$ is **arc-connected in $T$** if and only if every two points in $U$ are arc-connected in $T\,'$.

That is, $U$ is **arc-connected** if and only if:

- for every $x, y \in U$, there exists a continuous injection $f: \left[{0 \,.\,.\, 1}\right] \to S$ such that $f \left({0}\right) = x$ and $f \left({1}\right) = y$.

## Also known as

The term **arc-connected** can also be seen unhyphenated: **arc connected**.

Some sources also refer to this condition as **arcwise-connected** or **arcwise connected**, but the extra syllable does not appear to add to the understanding.

## Also see

- Results about
**arc-connected spaces**can be found here.