Definition:Arc-Connected/Points

Definition

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

Let $a, b \in S$ be such that there exists an arc from $a$ to $b$.

That is, there exists a continuous injection $f: \closedint 0 1 \to S$ such that $\map f 0 = a$ and $\map f 1 = b$.

Then $a$ and $b$ are arc-connected.

It is also declared that any point $a$ is arc-connected to itself.

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

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $4$: Connectedness