AI 中文总结
本文以塔拉格兰德紧集为对象研究二维不交拷贝性质及逐点商问题。通过特定构造得出相关结论，并给出反例。还分离出局部凸观察结果，表明塔拉格兰德紧集无经典逐点序列商空间，相关可度量化商问题仍待解决。
AI 中文摘要
我们重新审视塔拉格兰德的 CH 紧集，将其作为二维不交拷贝性质和逐点商问题的测试对象。二维不交拷贝性质（2DCP）是空间 \(C_{\operatorname{p}}(X)\) 存在无限维可度量化商空间的拓扑充分条件；近期工作探讨塔拉格兰德紧集是否具有此性质。在假设对于平稳余平稳 \(S\subseteq\omega_1\) 有 \(\diamondsuit(S)\) 的条件下，我们通过额外的对角化进行塔拉格兰德的逆极限构造。得到的紧集 \(T\) 保留了塔拉格兰德的结论：\(C(T)\) 是格罗滕迪克空间，弱星紧球 \(M_1(T)\) 不包含 \(\beta\omega\) 的拷贝，且 \(T\) 没有非平凡收敛序列。同时，\(T\) 不存在两个不交的非可度量化闭子空间同胚；因此 \(T\) 没有 2DCP 且不是局部齐性的。我们还给出一个 ZFC 下的完美紧空间例子，它具有 2DCP 但不是局部齐性的，且既不包含 \(\beta\omega\) 也不包含 \(2^\omega\)。最后，我们分离出一个一般的局部凸观察结果，类似于巴拿赫 - 加布里耶良关于约瑟夫森 - 尼森佐维格性质的理论，表明到 \((\ell_p)_{\operatorname{p}}\)（\(1\leqslant p<\infty\)）的逐点商空间迫使具有约瑟夫森 - 尼森佐维格性质。因此塔拉格兰德紧集没有经典的逐点序列商空间 \((c_0)_{\operatorname{p}}\)、\((\ell_p)_{\operatorname{p}}\) 或 \((\ell_\infty)_{\operatorname{p}}\)。这些 \(C_{\operatorname{p}}\) - 空间的完全可度量化商问题仍然开放。文中包含几个开放问题。
英文摘要
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 $S\subseteqω_1$, 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 $βω$, 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.
Comments20 pp
