On Weierstrass compact pseudometric spaces and a weak form of the axiom of choice
type: Item
instance of: scholarly article
title: On Weierstrass compact pseudometric spaces and a weak form of the axiom of choice
zbMATH Open document ID: 0959.03034
DOI: 10.1016/S0166-8641(99)00123-6
author: Kyriakos Keremedis
published in: Topology and its Applications
publication date: 2001-04-24T00:00:00Z
review text: \textit{H. L. Bentley} and \textit{H. Herrlich} [Topology Appl. 85, No. 1-3, 153-164 (1998; Zbl 0922.03068)] investigated in ZF (= Zermelo-Fraenkel set theory without the Axiom of Choice) the relations between various compactness notions for pseudometric spaces. The question, under which set-theoretical condition every Weierstrass-compact (i.e., every infinite subset has an accumulation point) pseudometric space is compact, remained open. In the paper under review the author provides an elegant solution for this problem by showing that in ZF the following conditions are equivalent: Every Weierstrass-compact pseudometric space is compact. The Countable Axiom of Multiple Choice holds. Every Weierstrass-compact pseudometric space is countably compact. Every compact pseudometric space contains a dense subset which is a countable union of finite sets. (In the above mentioned paper by Bentley and Herrlich it has been shown that the Axiom of Countable Choice holds iff every compact pseudometric space is separable).
reviewed by: Horst Herrlich
Mathematics Subject Classification ID: 03E25
Mathematics Subject Classification ID: 54D30
Mathematics Subject Classification ID: 54A35
zbMATH DE Number: 1538677
zbMATH Keywords: Axiom of Countable Multiple Choice
zbMATH Keywords: compact
zbMATH Keywords: Weierstrass-compact
zbMATH Keywords: pseudometric spaces
MaRDI profile type: Publication
Wikidata QID: Q114012791
full work available at URL: s0166-8641(99)00123-6
OpenAlex ID: W2080952817
description: scientific article; zbMATH DE number 1538677