QN-spaces, wQN-spaces and covering properties
type: Item
instance of: scholarly article
title: QN-spaces, wQN-spaces and covering properties
zbMATH Open document ID: 1117.54003
author: Jozef Haleš
author: Lev Bukovský
published in: Topology and its Applications
publication date: 2007-03-08T00:00:00Z
review text: Let \(X\) be an infinite Hausdorff space. \(X\) is an \({\mathcal S}^*\)-space if \({\mathrm{scl}} ({\mathrm{scl}} (A))={\mathrm{scl}}(A)\) for every subset \(A \subseteq X\) where \({\mathrm{scl}} (A)\) denotes the sequential closure of \(A\). \(X\) satisfies \((\alpha_1)\) if for any \(x \in X\) and any collection \(\{(x_{n,m})_m : n \in {\mathbb{N}} \}\) of sequences converging to \(x\), there is a sequence \((x_n)_n\) converging to \(x\) which contains all but finitely many members of every \((x_{n,m})_m\). If we only require that \((x_n)_n\) contains a member of \((x_{n,m})_m\) for infinitely many \(n\), then \(X\) is said to satisfy \((\alpha_4)\). If we can always find an unbounded non-decreasing sequence of natural numbers \((n_m)_m\) such that \((x_{n_m,m})_m\) converges to \(x\), then we say \(X\) satisfies \((\alpha_0)\). Properties \((\alpha_1)\) and \((\alpha_4)\) were introduced by Arkhangel'skiǐ and \((\alpha_0)\) by the authors. Clearly, \((\alpha_1)\) implies \((\alpha_0)\) which in turn implies \((\alpha_4)\). A sequence \((f_n)_n\) of real-valued functions on \(X\) converges quasinormally to a function \(f\) if there is a sequence \((\epsilon_n)_n\) of positive reals converging to \(0\) such that for all \(x \in X\), \(| f_n (x) - f(x) | < \epsilon_n\) holds for all but finitely many \(n\). Quasinormal convergence is stronger than pointwise convergence and weaker than uniform convergence. \(X\) is a QN-space if every sequence \((f_n)_n\) converging pointwise to \(0\) also converges quasinormally to \(0\). \(X\) is a wQN-space if every such \((f_n)_n\) contains a subsequence converging quasinormally to \(0\). \(C_p (X)\) denotes the space of real-valued functions on \(X\) with the topology of pointwise convergence. An infinite cover \({\mathcal U}\) of \(X\) is a \(\gamma\)-cover if every \(x \in X\) belongs to all but finitely many members of \({\mathcal U}\). A \(\gamma\)-cover \({\mathcal U}\) is shrinkable if there is a closed \(\gamma\)-cover …
Mathematics Subject Classification ID: 54A20
Mathematics Subject Classification ID: 54C30
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 54D20
zbMATH DE Number: 5131636
zbMATH Keywords: property \((\alpha_1)\)
zbMATH Keywords: property \((\alpha_4)\)
zbMATH Keywords: property \((\alpha_0)\)
zbMATH Keywords: pointwise convergence
zbMATH Keywords: quasinormal convergence
zbMATH Keywords: QN-space
zbMATH Keywords: wQN-space
zbMATH Keywords: \(\gamma\)-cover
zbMATH Keywords: covering property
MaRDI profile type: Publication
full work available at URL: j.topol.2006.09.008
OpenAlex ID: W2172281771
cites work: Q5184124
cites work: Q4018746
cites work: Q4828463
cites work: On Hurewicz properties.
cites work: Thin sets of harmonic analysis and infinite combinatorics
cites work: Spaces not distinguishing pointwise and quasinormal convergence of real functions
cites work: Spaces not distinguishing convergences of real-valued functions.
cites work: Q4190850
cites work: Q4150804
cites work: Q4305513
cites work: Some properties of C(X). I
cites work: Q4363975
cites work: Combinatorics of open covers. I: Ramsey theory
cites work: A sequential property of $\mathsf {C}_p(X)$ and a covering property of Hurewicz
cites work: \(C_p(X)\) and Arhangel'skiĭ's \(\alpha_i\)-spaces
cites work: Q4951594
cites work: The combinatorics of Borel covers
DOI: 10.1016/J.TOPOL.2006.09.008
Recommended article: Q3161719
Recommended article: \(Q_p\)-spaces
Recommended article: Q4650542
Recommended article: On wQN\(_*\) and wQN\(^*\) spaces
Recommended article: \(Q_p\) spaces -- a survey
Recommended article: Quasi-\(WCG\)-spaces
Recommended article: Q4942184
Recommended article: \(\mathcal{Q}_K\) spaces: a brief and selective survey
Recommended article: Q3692454
Recommended article: \(\mathcal{Q}\)-closure spaces
description: scientific article; zbMATH DE number 5131636