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 connected set.

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

Hence the result.