# Definition:Path-Connected/Topology/Set

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## Definition

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.