Talagrand compacta, 2DCP, and pointwise quotients

Abstract

We revisit Talagrand&#39;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&#39;s compactum has this property. Assuming $\diamondsuit(S)$ for a stationary co-stationary $S\subseteqω_1$, we carry out Talagrand&#39;s inverse-limit construction with additional diagonalisation. The resulting compactum $T$ keeps Talagrand&#39;s conclusions: $C(T)$ is Grothendieck, the weak-star compact ball $M_1(T)$ contains no copy of $βω$, 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 $βω$ nor $2^ω$. 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.