Ultraconnected Space is not necessarily Arc-Connected

Theorem

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

Then $T$ is not necessarily arc-connected.

Proof

Let $T$ be an excluded point space.

From Excluded Point Space is Ultraconnected, $T$ is an ultraconnected space.

From Excluded Point Space is not Arc-Connected, $T$ is not arc-connected.

Hence the result.

$\blacksquare$

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