Definition:Arc-Connected/Topological Space

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Definition

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 $\map f 1 = y$.


Also known as

The term arc-connected can also be seen unhyphenated: arc connected.

Some sources also refer to this condition as:

arcwise-connected
arcwise connected
arc-wise connected

but the extra syllable does not appear to add to the understanding.


Also see

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


Sources