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

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

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

Then $U$ is a path-connected set in $T$ if and only if every two points in $U$ are path-connected in $T\,'$.

That is, $U$ is a path-connected set in $T$ if and only if:

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

Also known as

Some sources refer to this as a path-connected subset of $T$, but strictly speaking the subset nature of $U$ is of the underlying set $S$, not of $T$.

Thus, on $\mathsf{Pr} \infty \mathsf{fWiki}$, path-connected set is preferred, which is consistent with the concepts open set, closed set, connected set, and so on.

Some sources do not hyphenate, but instead report this as path connected.

Also see

  • Results about path-connected sets can be found here.