On a topological choice principle by Murray Bell
type: Item
instance of: scholarly article
title: On a topological choice principle by Murray Bell
zbMATH Open document ID: 1275.03141
author: Eleftherios Tachtsis
author: Paul Howard
published in: Topology and its Applications
publication date: 2013-02-19T00:00:00Z
review text: In what follows, \(\mathbf{ZF}\) is Zermelo-Fraenkel set theory, \(\mathbf{AC}\) denotes the axiom of choice, \(\mathbf{AC}^{\aleph_0}\) denotes the axiom of countable choice (which is the restriction of \(\mathbf{AC}\) to countable families of non-empty sets) and \(\mathbf{BPI}\) denotes the Boolean prime ideal theorem. In the paper under review, the authors investigate the consistency strength within \(\mathbf{ZF}\) of the following statement, attributed to Murray Bell: \(\mathbf{(C)}\): For every family \(\{A_i: i \in I\}\) of non-empty sets, there exists a family \(\{\tau_i: i \in I\}\) such that for every \(i \in I\), \(\tau_i\) is a compact \(T_2\) topology on \(A_i\). One has \(\mathbf{(C)} + \mathbf{BPI} \Leftrightarrow \mathbf{AC}\); this result is attributed in the paper to \textit{H. Herrlich} and \textit{K. Keremedis} [Topol. Appl. 158, No. 17, 2279--2286 (2011; Zbl 1231.03039)]. The following questions are still unsolved, as the authors themselves remark: are \(\mathbf{(C)}\) and \(\mathbf{AC}\) equivalent ? If not, which choice principles, if any, are equivalent to \(\mathbf{(C)}\)? The pursuit of answers to both questions constitute what the authors state as Bell's Problem. Several results regarding \(\mathbf{(C)}\) are established in the paper. Among many others, the following statements are proved: {\parindent=0.6cm\begin{itemize}\item[(i)] \(\mathbf{(C)}\) + ``For every set \(X\), every countable filter base on \(X\) can be extended to an ultrafilter on \(X\)''\, implies \(\mathbf{AC}^{\aleph_0}\). \item[(ii)] \(\mathbf{(C)}\) restricted to countable families of non-empty sets + ``For every set \(X\), every countable filter base on \(X\) on \(X\) can be extended to an ultrafilter on \(X\)''\,is equivalent to \(\mathbf{AC}^{\aleph_0}\) + ``There exists a free ultrafilter on \(\omega\)''. \item[(iii)] \(\mathbf{(C)}\) restricted to countable families of non-empty sets does not imply ``There exists a free ultrafilter on \(\omega\)''\, in \(\mathbf{ZF}…
reviewed by: Samuel Gomes da Silva
Mathematics Subject Classification ID: 03E25
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 54D10
Mathematics Subject Classification ID: 54D30
zbMATH DE Number: 6137379
zbMATH Keywords: axiom of choice
zbMATH Keywords: weak forms of the axiom of choice
zbMATH Keywords: compact \(T_{2}\) topological space
zbMATH Keywords: Fraenkel-Mostowski permutation models of ZFA
zbMATH Keywords: symmetric models of ZF
zbMATH Keywords: consistency strength
MaRDI profile type: Publication
full work available at URL: j.topol.2012.11.008
OpenAlex ID: W2038294373
DOI: 10.1016/J.TOPOL.2012.11.008
description: scientific article; zbMATH DE number 6137379