2026/07/07 by Tomasz Kania, Jerzy Kąkol · 1 voice
Mathematics · #math.GN
arxiv published 2026/07/07 · arxiv updated 2026/07/10
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 Cp(X); recent work asks whether Talagrand's compactum has this property. Assuming \diamondsuit(S) for a stationary co-stationary S⊆ω1, 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 M1(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 2ω. 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 (ℓp)p, 1\leqslant p<∞, force the Josefson--Nissenzweig property. Consequently Talagrand compacta have no classical pointwise sequence quotients (c0)p, (ℓp)p, or (ℓ_∞)p. The full metrisable quotient problem for these Cp-spaces remains open. Several open problems are included.