Definition:Path-Connected/Topology/Topological Space

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Definition

Let $T = \struct {S, \tau}$ 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 $f: \closedint 0 1 \to S$ such that:
$\map f 0 = x$
and:
$\map f 1 = y$


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.


Also see

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


Sources