Definition:Arc-Connected/Subset

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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.