Existence of Connected Space which is Totally Pathwise Disconnected

Theorem

There exists at least one example of a topological space which is both connected and totally pathwise disconnected.

Proof

Let $T$ be Gustin's sequence space.

From Gustin's Sequence Space is Connected, $T$ is a connected space.

From Gustin's Sequence Space is Totally Pathwise Disconnected, $T$ is a totally pathwise disconnected space.

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: Disconnectedness