A classification of the cofinal structures of precompacta
type: Item
instance of: scholarly article
title: A classification of the cofinal structures of precompacta
zbMATH Open document ID: 1452.03101
author: Aviv Eshed
author: M. Vicenta Ferrer
author: P. Szewczak
author: Boaz Tsaban
author: Salvador Hernández
published in: Annals of Pure and Applied Logic
publication date: 2020-06-02T00:00:00Z
full work available at URL: https://arxiv.org/abs/1607.07669
full work available at URL: http://hdl.handle.net/10234/189225
review text: The purpose of this paper is to classify the families of precompact (totally bounded) sets in general metric spaces up to cofinal equivalence, where two partial orders are cofinally equivalent if each admits a monotone map onto a cofinal subset of the other. Note that this then also classifies the families of compact sets of general \textit{complete} metric spaces. Since `all' cardinal functions coincide in the class of metric spaces and are equal to the density the latter features prominently in the results. Indeed, if \(X\) has a locally compact completion then its family of precompact sets is equivalent to the family of finite subsets of the cardinal \(d(X)\) (provided \(X\) itself is not precompact). In the more general case one obtains the product of the latter partial order and a subset of the set functions from \(\mathbb{N}\) to \([ld(\tilde X)]^{<\aleph_0}\), where \(ld(\tilde X)\) is the local density of the completion of \(X\); depending on whether the latter is attained or not one has the full set of functions or just the set of bounded functions. These results are applied in the computation of the cofinalities of the families in question and of the weight and character of topological groups of the form \(C(X,G)\) where the latter carries the compact-open topology.
reviewed by: K. P. Hart
Mathematics Subject Classification ID: 03E15
Mathematics Subject Classification ID: 03E04
Mathematics Subject Classification ID: 03E17
Mathematics Subject Classification ID: 06A07
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 54A25
zbMATH DE Number: 7207189
zbMATH Keywords: cofinal equivalence
zbMATH Keywords: Tukey equivalence
zbMATH Keywords: compact sets
zbMATH Keywords: precompact sets
zbMATH Keywords: compact-open topology
MaRDI profile type: Publication
OpenAlex ID: W3015529024
arXiv ID: 1607.07669
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DOI: 10.1016/J.APAL.2020.102810
description: scientific article