On a topological choice principle by Murray Bell (Q1935840)

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 \(\mathbf{ZF}\). \item[(iv)] \(\mathbf{(C)}\) + ``The axiom of choice for countable families of non-empty sets of reals\,''\,implies ``There exists a non-Lebesgue measurable set of reals''. \end{itemize}} The paper ends by posing a number of questions, for instance: does \(\mathbf{(C)}\) imply ``For a product of a countable family of compact \(T_2\) spaces, canonical projections are closed mappings''\, ? Reviewer's addendum: As already remarked, the topological choice principle \(\mathbf{(C)}\) is attributed in the paper to Murray Bell. However, such statement was considered way back in the 50s by \textit{E. Farah} [Bol. Soc. Mat. Sao Paulo 10, 1--65 (1958; Zbl 0116.00903)], where he also established the equivalence between \(\mathbf{AC}\) and \(\mathbf{(C)}\) + \(\mathbf{BPI}\); as previously commented, such equivalence is attributed in the paper under review to Herrlich and Keremedis. In Farah's paper [loc. cit.], \(\mathbf{(C)}\) + \(\mathbf{BPI}\) (in fact, \(\mathbf{(C)}\) + \(\mathbf{UT}\), where \(\mathbf{UT}\) denotes the ultrafilter theorem -- which is equivalent to \(\mathbf{BPI}\) and states that every filter (on any set) can be extended to an ultrafilter) is statement \(\mathrm P_{\mathrm{VII}}\), one of seven statements shown pairwise equivalent (with \(\mathrm P_{\mathrm I}\) being \(\mathbf{AC}\)).