Sequential order of product of Fréchet spaces
type: Item
instance of: scholarly article
title: Sequential order of product of Fréchet spaces
zbMATH Open document ID: 0854.54024
DOI: 10.1016/0166-8641(95)00099-2
author: T. Nogura
author: Alexander Y. Shibakov
published in: Topology and its Applications
publication date: 1996-09-04T00:00:00Z
review text: For a subset \(A\) of a space, \([A ]^{\text{seq}}\) denotes the sequential closure of \(X\), i.e., the set of limits of convergent sequences in \(A\). For a sequential space \(X\), we define \(\text{so} (X)\), the sequential order of \(X\), by \(\text{so} (X)= \min \{\alpha\in \omega_1 + 1: \text{cl } A=[ A]^\alpha\) for every \(A\subset X\}\), where \([A ]^0 =A\), \([A ]^{\alpha+ 1}= [[ A]^\alpha ]^{\text{seq}}\), and \([A ]^\alpha= \bigcup \{[ A]^\beta: \beta< \alpha\}\) for a limit \(\alpha\). A space \(X\) is strongly Fréchet if whenever \(x\in \text{cl } A_n\) \((n\in \mathbb{N})\), here \(A_{n+1} \subset A_n\), there exists a sequence \(\{x_n: n\in \mathbb{N}\}\) converging to \(x\) with \(x_n\in A_n\). In this paper, assuming CH, for any given \(\alpha\leq \omega_1\) there exist strongly Fréchet spaces \(X\) and \(Y\) such that \(X \times Y\) is sequential, and \(\text{so} (X\times Y)= \alpha\), thus answering a problem in the authors' previous paper [ibid. 65, 271-285 (1995; Zbl 0837.54021)].
Mathematics Subject Classification ID: 54D55
Mathematics Subject Classification ID: 54B10
Mathematics Subject Classification ID: 54A20
Mathematics Subject Classification ID: 54D50
Mathematics Subject Classification ID: 54G15
zbMATH DE Number: 896619
zbMATH Keywords: sequential closure
zbMATH Keywords: sequential space
zbMATH Keywords: sequential order
zbMATH Keywords: strongly Fréchet spaces
reviewed by: Y. Tanaka
MaRDI profile type: Publication
full work available at URL: 0166-8641(95)00099-2
OpenAlex ID: W2017603064
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description: scientific article; zbMATH DE number 896619