Category:Definitions/Path-Connected Spaces

This category contains definitions related to Path-Connected Spaces.
Related results can be found in Category:Path-Connected Spaces.

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:

$\map f 1 = y$