# Talagrand compacta, 2DCP, and pointwise quotients

Source: [Talagrand compacta, 2DCP, and pointwise quotients](<https://www.sota2.com/research/paper/6a4f2aeb32fb937314875abe>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2026-07-14T05:00:01.588Z

Authors: Tomasz Kania, Jerzy K\\k{a}kol

Published: 2026-07-07T21:14:22.000Z

## Abstract

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.

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2607.06808>)

[PDF](<https://arxiv.org/pdf/2607.06808.pdf>)