Definition:Path-Connected/Topology/Topological Space

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Definition

Let $T = \left({S, \tau}\right)$ be a topological space.


Then $T$ is a path-connected space if and only if $S$ is a path-connected set of $T$.


That is, $T$ is a path-connected space if and only if:

for every $x, y \in S$, there exists a continuous mapping $f: \left[{0 \,.\,.\, 1}\right] \to S$ such that $f \left({0}\right) = x$ and $f \left({1}\right) = y$.


Also known as

Some sources do not hyphenate, but instead report this as path connected.


Also see

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


Sources