Definition:Path-Connected/Metric Space/Subset
< Definition:Path-Connected | Metric Space(Redirected from Definition:Path-Connected Metric Subspace)
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Definition
Let $M = \struct {A, d}$ be a metric space.
Let $S \subseteq A$ be a subset of $M$.
Then $S$ is path-connected (in $M$) if and only if:
- $\forall m, n \in S: \exists f: \closedint 0 1 \to S: \map f 0 = m, \map f 1 = n$
where $f$ is a continuous mapping.
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.
Sources
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{III}$: Metric Spaces: Path-Connectedness