Definition:Arc-Connected/Subset

Definition

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

Let $U \subseteq S$ be a subset of $T$.

Let $T' = \struct {U, \tau_U}$ be the subspace of $T$ induced by $U$.

Then $U$ is arc-connected in $T$ if and only if every two points in $U$ are arc-connected in $T'$.

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

for every $x, y \in U$, there exists a continuous injection $f: \closedint 0 1 \to S$ 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.