Equalities for the extent of infinite products and Σ-products
Equalities for the extent of infinite products and Σ-products
复制标题
无限乘积和 Σ-乘积范围的等式
DOI：
10.1016/j.topol.2021.107946
复制
发表时间：
2022
影响因子：
0.6
通讯作者：
Yajima Yukinobu
中科院分区：
文献类型：
--
作者：
Hirata Yasushi;Usuba Toshimichi;Yajima Yukinobu
对于空间X，设e（X）= ω sup {|D|：D是X}中的一个闭离散子集，称为X的范围。本文讨论了以下两个问题：（1）对于乘积空间X=<$λ∈ Λ X λ，当e（X）=| Λ| {e（X λ）：λ∈ Λ}？(2)对于空间X λ，λ∈ Λ的一个乘积ε，当e（ε）= sup {e（X λ）：λ∈ Λ}？我们证明了当每个X λ是严格p-空间或强p-空间时，以及在第一个问题的情况下，当指标集Λ的基数小于第一个弱不可达时，这些问题中的等式成立。对于半层空间，我们证明了这些等式的一个稍弱的形式成立。
For a space X, let e (X)= ω⋅ sup {| D|: D is a closed discrete subset in X}, which is called the extent of X. Here we deal with the following two questions:(1) For a product space X=∏ λ∈ Λ X λ, when is e (X)=| Λ|⋅ sup {e (X λ): λ∈ Λ}?(2) For a Σ-product Σ of spaces X λ, λ∈ Λ, when is e (Σ)= sup {e (X λ): λ∈ Λ}? We show that the equalities in these questions hold if each X λ is a strict p-space or a strong Σ-space and, in the case of the first question, if the cardinality of the index set Λ is less than the first weakly inaccessible. For semi-stratifiable spaces, we show that a slightly weaker form of these equalities holds.