On multisequences and their application to products of sequential spaces
type: Item
instance of: scholarly article
zbMATH Open document ID: 0957.54011
author: Saliou Sitou
publication date: 2001-03-13T00:00:00Z
full work available at URL: https://eudml.org/doc/32163
Mathematics Subject Classification ID: 54D55
Mathematics Subject Classification ID: 54B10
zbMATH DE Number: 1575491
zbMATH Keywords: sequential topology
zbMATH Keywords: sequential order
zbMATH Keywords: multisequence product of sequential spaces
MaRDI profile type: Publication
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cites work: Products of Sequential Spaces
title: On multisequences and their application to products of sequential spaces
published in: Mathematica Slovaca
review text: Let \(X\) be a topological space and let \(\operatorname {cl}\) be the closure operator in \(X\). A sequential adherence of \(A\subset X\) is the set of all limits of convergent sequences ranging in \(A\). The corresponding sequential adherence operator \(\operatorname {cl}_{\text{seq}}\) need not be idempotent and can be iterated up to \(\omega_1\) times -- this way we get an idempotent operator \(\operatorname {cl}_{T\operatorname {seq}}\) called the sequential closure of \(A\). If \(\operatorname {cl}_{\operatorname {seq}}=\operatorname {cl}\), then \(X\) is a Fréchet space and if \(\operatorname {cl}_ {T\operatorname {seq}}=\operatorname {cl}\), then \(X\) is a sequential space. The sequential order \(\sigma (X)\) of \(X\) is at most \(\omega_1\) and it is roughly ``how many times we have to iterate \(\operatorname {cl}_{\operatorname {seq}}\) to get \(\operatorname {cl}_{T\operatorname {cl}}\)''. NEWLINENEWLINENEWLINEThe paper under review deals with sequential properties of products. The main result (Theorem 3.3): If \(X\) is a sequential space and \(Y\) is a sequential regular locally countably compact space, then \(X\times Y\) is sequential and \(\sigma (X\times Y)\leqq \sigma (X) + \sigma (Y)\). The result generalizes several known theorems about products by A. V. Arkhangel'skij, E. Michael, T. Nogura and A. Shibakov. NEWLINENEWLINENEWLINEFrom the technical viewpoint, all is based on the convergence of multisequences -- a useful but rather technical construction. If \(x\in \operatorname {cl}_{T\operatorname {seq}} A\), then there is a sequence \((x(n))\) of points ``sequentially closer to \(A\) than \(x\)'', each \(x(n)\) is a limit of a sequence \((x(n,m))\) of points ``sequentially closer to \(A\) than \(x(n)\)'', an so on. Due to the compactness of \(\omega_1+1\) the ``fibres'' are finite and we always end up in \(A\) after finitely many steps. Hence multisequences ``live within the set of finite sequences of natural numbers'' and their convergence is de…
reviewed by: Roman Frič
description: scientific article