Arc-Connected Space is Path-Connected

Theorem

Let $T = \struct {S, \tau}$ be a topological space which is arc-connected.

Then $T$ is path-connected.

Proof

Let $T = \struct {S, \tau}$ be arc-connected.

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

As $f$ is a continuous injection, it is also simply a continuous mapping.

The result follows from the definition of path-connectedness.

$\blacksquare$

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