On Hurewicz properties.
type: Item
instance of: scholarly article
title: On Hurewicz properties.
zbMATH Open document ID: 1056.54024
DOI: 10.1016/S0166-8641(02)00364-4
author: Lev Bukovský
author: Jozef Haleš
published in: Topology and its Applications
publication date: 2003-08-07T00:00:00Z
review text: \textit{W. Hurewicz} in 1925 introduced several properties of a topological space \(X\) concerning sequences of real functions and open covers, e.g., Property \(H^{**}\): For any sequence \(\{f_n\}^\infty_{n= 0}\) of continuous function from \(X\) into \(\mathbb{R}\) the family of sequences in \(\mathbb{R}\) \(\{\{t_n(x)\}_{n=0}^\infty |x\in X\}\) is bounded. Property \(E^{**}\): for every sequence \(\{U_n\}^\infty_{n =0}\) of open covers of \(X\) there exist finite susets \(V_n\subseteq U_n\) such that \(\{\bigcup V_n|n\in\omega\}\) is a \(\gamma\)-cover of \(X\) is a finite cover. The authors of the present paper investigate relations between these properties and several generalizations. Their main result is the following: Consider the property of being an \(mQN\)-space, introduced by \textit{L. Bukovský}, \textit{I. Reclaw} and \textit{M. Repický} [Topology Appl. 41, 25--40 (1991; Zbl 0768.54025) and ibid. 112, 13--40 (2001; Zbl 1067.54028)]: \(X\) is called an \(mQN\)-space if every sequence \(f_n\to 0\) of contiuous real-valued functions converges quasi-normally. The authors prove: Theorem 6: Every \(mQN\)-space possesses property \(H^{**}\).
Mathematics Subject Classification ID: 54D20
Mathematics Subject Classification ID: 54A20
Mathematics Subject Classification ID: 03E75
Mathematics Subject Classification ID: 54C50
Mathematics Subject Classification ID: 26A99
Mathematics Subject Classification ID: 54C30
Mathematics Subject Classification ID: 03E05
zbMATH DE Number: 1961617
zbMATH Keywords: Eventual dominating
zbMATH Keywords: Hurewicz property
zbMATH Keywords: Lindelöf property
zbMATH Keywords: Monotonic
zbMATH Keywords: sequence selection property
zbMATH Keywords: mQN-space
MaRDI profile type: Publication
cites work: Spaces not distinguishing pointwise and quasinormal convergence of real functions
cites work: Spaces not distinguishing convergences of real-valued functions.
cites work: Q4150804
cites work: Q5186348
cites work: \(\gamma\)-sets and other singular sets of real numbers
cites work: Some properties of C(X). I
cites work: The combinatorics of open covers. II
cites work: On some properties of Hurewicz, Menger, and Rothberger
cites work: A sequential property of $\mathsf {C}_p(X)$ and a covering property of Hurewicz
cites work: Q4951594
cites work: Combinatorics of open covers. I: Ramsey theory
cites work: Q4164026
cites work: Q3998752
description: scientific article; zbMATH DE number 1961617