# Definition:Path-Connected/Topology/Set

## Definition

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:
- $\map f 0 = x$

- and:
- $\map f 1 = y$

## 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**.