# Definition:Path-Connected/Metric Space

< Definition:Path-Connected(Redirected from Definition:Path-Connected Metric Space)

## Definition

Let $M = \left({A, d}\right)$ be a metric space.

$M$ is defined as **path-connected** iff:

- $\forall m, n \in A: \exists f: \left[{0 \,.\,.\, 1}\right] \to A: f \left({0}\right) = m, f \left({1}\right) = n$

where $f$ is a continuous mapping.

### Subset of Metric Space

Let $M = \left({A, d}\right)$ be a metric space.

Let $S \subseteq A$ be a subset of $M$.

Then $S$ is **path-connected (in $M$)** if and only if:

- $\forall m, n \in S: \exists f: \left[{0 \,.\,.\, 1}\right] \to S: f \left({0}\right) = m, f \left({1}\right) = n$

where $f$ is a continuous mapping.

## Also see

- Results about
**path-connected spaces**can be found here.

## Sources

- 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): $\text{III}$: Path-Connectedness