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 $T = \left({S, \tau}\right)$ 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:

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