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Let $T = \struct {S, \tau}$ be a topological space.

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

Let $T' = \struct {U, \tau_U}$ 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 $f: \closedint 0 1 \to U$ such that:
$\map f 0 = x$
$\map f 1 = y$

Also known as

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

Some sources use path-wise connected

Some sources use the term arc-connected or arc-wise connected, but this normally has a more precise meaning.

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.

Also see

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