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C⁎-, C- and P-embedded subsets in products and undecidability of a certain property on Nω1
Topology and its Applications ( IF 0.6 ) Pub Date : 2020-09-01 , DOI: 10.1016/j.topol.2020.107350
Yasushi Hirata , Yukinobu Yajima

Abstract Let X = ∏ α ∈ ω 1 X α be a product of ω 1 many separable metric spaces. It is proved that every C ⁎ -embedded subset in X is C-embedded in X under a certain set-theoretic assumption. Combining a result of E. Pol and R. Pol for N 2 ω , it is undecidable in ZFC that there is a (closed) C ⁎ -embedded but not C-embedded subset in N ω 1 . It is also proved in ZFC that every C-embedded subset in X is P-embedded in X. Next, we discuss when C ⁎ - or C-embedding implies P-embedding in products of generalized metrizable spaces, such as M-spaces, Σ-spaces and semi-stratifiable spaces.

中文翻译:

乘积中的 C⁎-、C- 和 P-嵌入子集以及 Nω1 上某个属性的不可判定性

摘要 令X = ∏ α ∈ ω 1 X α 是ω 1 许多可分度量空间的乘积。证明了在某个集合论假设下,X 中的每个 C ⁎ 嵌入子集都是 C 嵌入 X 中的。结合N 2 ω 的E. Pol 和R. Pol 的结果,在ZFC 中不可判定在N ω 1 中存在(封闭)C ⁎ 嵌入但不存在C 嵌入子集。在 ZFC 中也证明了 X 中每个 C 嵌入的子集都是 P 嵌入 X。 接下来,我们讨论何时 C ⁎ - 或 C 嵌入意味着 P 嵌入在广义可度量空间的乘积中,例如 M 空间, Σ-空间和半分层空间。
更新日期:2020-09-01
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