Talagrand compacta, 2DCP, and pointwise quotients

We revisit Talagrand's CH compactum as a test object for the two-disjoint-copies property and for pointwise quotient questions.

The two-disjoint-copies property, or 2DCP, is a topological sufficient condition for the existence of infinite-dimensional metrisable quotients of spaces $C_{\operatorname{p}}(X)$; recent work asks whether Talagrand's compactum has this property.

Assuming $\diamondsuit(S)$ for a stationary co-stationary $S\subseteq\omega_1$, we carry out Talagrand's inverse-limit construction with additional diagonalisation.

The resulting compactum $T$ keeps Talagrand's conclusions: $C(T)$ is Grothendieck, the weak-star compact ball $M_1(T)$ contains no copy of $\beta\omega$, and $T$ has no non-trivial convergent sequences.

At the same time, no two disjoint non-metrisable closed subspaces of $T$ are homeomorphic; hence $T$ has no 2DCP and is not locally homogeneous.

We also give a ZFC example of a perfect compact space with 2DCP which is not locally homogeneous and contains neither $\beta\omega$ nor $2^\omega$.

Finally, we isolate a general locally convex observation, in the spirit of the Banakh--Gabriyelyan theory of the Josefson--Nissenzweig property, showing that pointwise quotients onto $(\ell_p)_{\operatorname{p}}$, $1\leqslant p<\infty$, force the Josefson--Nissenzweig property.

Consequently Talagrand compacta have no classical pointwise sequence quotients $(c_0)_{\operatorname{p}}$, $(\ell_p)_{\operatorname{p}}$, or $(\ell_\infty)_{\operatorname{p}}$.

The full metrisable quotient problem for these $C_{\operatorname{p}}$-spaces remains open.

Several open problems are included.