Definition:Path-Connected/Metric Space

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Definition

Let $M = \struct {A, d}$ be a metric space.

$M$ is defined as path-connected if and only if:

$\forall m, n \in A: \exists f: \closedint 0 1 \to A: \map f 0 = m, \map f 1 = n$

where $f$ is a continuous mapping.


Subset of Metric Space

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 see

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


Sources