Uniform Eberlein compactifications of metrizable spaces
type: Item
instance of: scholarly article
title: Uniform Eberlein compactifications of metrizable spaces
zbMATH Open document ID: 1257.54028
author: Taras Banakh
author: Arkady Leiderman
published in: Topology and its Applications
publication date: 2012-04-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/1012.0920
reviewed by: Rodrigo R. Dias
Mathematics Subject Classification ID: 54D35
Mathematics Subject Classification ID: 54G12
Mathematics Subject Classification ID: 54D30
Mathematics Subject Classification ID: 54D20
zbMATH DE Number: 6024146
zbMATH Keywords: scattered space
zbMATH Keywords: metrizable space
zbMATH Keywords: scattered compactification
zbMATH Keywords: hereditarily paracompact space
zbMATH Keywords: uniform Eberlein compact space
MaRDI profile type: Publication
OpenAlex ID: W2017745476
arXiv ID: 1012.0920
cites work: Some remarks on Eberlein compacts
cites work: Q4002497
cites work: Embedding weakly compact sets into Hilbert space
cites work: The dimension of metrizable subspaces of Eberlein compacta and Eberlein compactifications of metrizable spaces
cites work: Q3994935
cites work: On Eberlein compactifications of metrizable spaces
cites work: Q3711330
cites work: Q4150322
cites work: What is a non-metrizable analog of metrizable compacta? I
cites work: Q3914065
cites work: Q4107619
cites work: Q5549620
DOI: 10.1016/J.TOPOL.2011.06.060
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review text: In this paper, the authors investigate uniform Eberlein compactifications of metrizable spaces.NEWLINENEWLINEThey start by giving a short proof that every metrizable space has a uniform Eberlein compactification, which is a strengthening of a result of [\textit{A. V. Arkhangel'skii}, Topological function spaces. Mathematics and its Applications, Soviet Series, 78. Dordrecht: Kluwer Academic Publishers (1992; Zbl 0758.46026)]. As is noted in the paper, this first result was also obtained independently by \textit{B. A. Pasynkov} [Topology Appl. 159, No. 7, 1750--1760 (2012; Zbl 1252.54020)]; although Pasynkov's result yields such a compactification with stronger properties, the proof presented here is more straightforward. The authors also show that, if (and only if) the metrizable space has size not exceeding the continuum, one can obtain a first countable such compactification.NEWLINENEWLINEThe last two thirds of the paper are dedicated to the study of scattered uniform Eberlein compactifications of scattered metrizable spaces. The authors characterize the class of compact scattered hereditarily paracompact Hausdorff spaces as the smallest class \(\mathcal A\) that contains the finite discrete spaces and is closed under taking the Alexandroff compactifications of topological sums of elements of \(\mathcal A\). It is then proved that every space in the class \(\mathcal A\) is a uniform Eberlein compact, and that every scattered metrizable space has a compactification that belongs to \(\mathcal A\).
description: scientific article; zbMATH DE number 6024146