Definition:Path-Connected/Metric Space

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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