Existence of Biconnected Set without Dispersion Point

Theorem

There exists at least one example of a biconnected set which does not have a dispersion point.

Proof

Let $T$ be a Miller's biconnected set.

From Miller's Biconnected Set is Biconnected, $T$ is a biconnected set.

From Miller's Biconnected Set is has no Dispersion Point, $T$ does not have a dispersion point.

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: Biconnectedness and Continua