Category:Definitions/Arc-Connected Spaces

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

Let $T = \struct {S, \tau}$ be a topological space.

Then $T$ is arc-connected if and only if every two points in $T$ are arc-connected in $T$.

That is, $T$ is arc-connected if and only if:

$\forall x, y \in S: \exists$ a continuous injection $f: \closedint 0 1 \to X$ such that $\map f 0 = x$ and $\map f 1 = y$.