Sequence selection principles for quasi-normal convergence
type: Item
instance of: scholarly article
title: Sequence selection principles for quasi-normal convergence
zbMATH Open document ID: 1233.54001
author: Lev Bukovský
author: Jaroslav Šupina
published in: Topology and its Applications
publication date: 2011-12-06T00:00:00Z
review text: The authors introduce and investigate the following sequence selection principles for quasi-normal and discrete convergence. Let \(A\), \(B\) denote one of the following types of convergence: \(P\)-pointwise; \(Q\)-quasi-normal; \(D\)-discrete. A topological space \(X\) satisfies the sequence selection principle \(ASB\) if for any functions \(f,f_nf^n_m:X\to \mathbb{R}\), \(n,m\in\omega\), with: (a)~\(f_n\) \(A\)-converges to \(f\) on \(X\); (b)~for every \(n\), \(f^n_m\) \(A\)-converges to \(f_n\); (c)~all \(f^n_m\) are continuous, there exists an unbounded \(\beta:\omega\to\omega\) such that \(f^n_{\beta(n)}\) \(B\)-converges to \(f\) on \(X\). Using this notion the authors obtain new and interesting results on QN-spaces. (Recall that a topological space \(X\) is a QN-space if each sequence of continuous real valued functions converging to zero on \(X\) converges to zero quasi-normally as well.) The main results: (1)~any QN-space satisfies the principle \(QSQ\) (Theorem~3); and (2)~if \(X\) is a normal topological space satisfying \(QSQ\), then every \(\Delta^0_2\)-measurable function from \(X\) into \([0,1]\) is a quasi-normal limit of a sequence of continuous functions, therefore also a discrete limit of such a sequence. These results allow to generalize some results of I. Recław, and B.~Tsaban and L.~Zdomskyy.
reviewed by: Tomasz Natkaniec
Mathematics Subject Classification ID: 54A20
Mathematics Subject Classification ID: 54C30
Mathematics Subject Classification ID: 54G99
zbMATH DE Number: 5986565
zbMATH Keywords: QN-space
zbMATH Keywords: Borel measurable function
zbMATH Keywords: Borel image
zbMATH Keywords: Baire space
zbMATH Keywords: selection principles for sequences of real functions
zbMATH Keywords: pointwise convergence
zbMATH Keywords: discrete convergence
zbMATH Keywords: quasi-normal convergence.
MaRDI profile type: Publication
full work available at URL: j.topol.2011.09.034
OpenAlex ID: W2058636761
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DOI: 10.1016/J.TOPOL.2011.09.034
description: scientific article; zbMATH DE number 5986565