Definition:Path-Connected/Metric Space/Subset

From ProofWiki
Jump to navigation Jump to search

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