Compact and Loeb Hausdorff spaces in ZF and the axiom of choice for families of finite sets (Q2888625)

A topological space \((X,T)\) is called a Loeb space if the family of all non-empty closed subsets of \(X\) has a choice function.NEWLINENEWLINELet \(\{(X_i,T_i):i\in I\}\) be a family of topological spaces and let \(X\) be their Tychonoff product. A closed set \(F\) of \(X\) is called restricted closed if there is a finite subset \(Q_F\) of \(I\) such that \(F=V\times\prod\limits_{i\in Q_F^{c}}X_i\) with \(V\subset Y_{Q_{F}}=\prod\limits_{i\in Q_F}X_i\) a closed set. The set \(Q_F\) is called a support of \(F\). According to Lemma 3.4 of the paper, to each non-empty proper restricted closed set \(F\) of \(X\), there corresponds a smallest (w.r.t. inclusion) support, which is called minimal support of \(F\). The Tychonoff product \(X\) is called \(R\)-Loeb if the family of all non-empty restricted closed subsets of \(X\) has a choice functionNEWLINENEWLINEIn the paper under review, the author studies the productivity of the Loeb notion in set theory without choice. In particular, he investigates the set-theoretic strength of the statements ``For every set \(X\), \(2^X\) is Loeb'', ``For every set \(X\), \(2^X\) is \(R\)-Loeb'', where in both statements, \(2\) is the discrete two-element space \(\{0,1\}\) and \(2^X\) is endowed with the product topology, and ``Tychonoff products of Loeb Hausdorff spaces are Loeb''.NEWLINENEWLINENote that the Boolean prime ideal theorem (BPI: Every non-trivial Boolean algebra has a prime ideal) implies ``For every set \(X\), \(2^X\) is Loeb'' (cf. [\textit{K. Keremedis} and \textit{E. Tachtsis}, ``On Loeb and weakly Loeb Hausdorff spaces'', Sci. Math. Jpn. 53, No. 2, 247--251 (2001; Zbl 0982.54001)]).NEWLINENEWLINEThe author establishes the subsequent results:NEWLINENEWLINELet \(X\) be any set. Then the following principles are pairwise equivalent:NEWLINENEWLINEAC\(^{\mathrm{fin}(X)}\), i.e., \([X]^{<\omega}\setminus\{\emptyset\}\) (= the set of all non-empty finite subsets of \(X\)) has a choice function.NEWLINENEWLINEAC\(^{\mathrm{fin}([X]^{<\omega})}\).NEWLINENEWLINEAC\(^{\mathrm{fin}(\mathrm{Fn}(X,2))}\), where \(\mathrm{Fn}(X,2)\) is the set of all finite partial functions from \(X\) into \(2\).NEWLINENEWLINE\(2^X\) is \(R\)-Loeb.NEWLINENEWLINEThe Tychonoff product \(\prod_{A\in [X]^{<\omega}\setminus\{\emptyset\}}A\), where each \(A\in [X]^{<\omega}\setminus\{\emptyset\}\) has the discrete topology, is \(R\)-Loeb. NEWLINENEWLINE{1)} AC\(_{\mathrm{fin}}\) (Every family of non-empty finite sets has a choice function) iff ``For every set \(X\), \(2^X\) is \(R\)-Loeb''. NEWLINENEWLINE{2)} ``For every set \(X\), \(2^X\) is Loeb'' and ``Every set can be expressed as a well-ordered union of well-orderable sets'' implies BPI. NEWLINENEWLINE{3)} ``For every set \(X\), \(2^X\) is Loeb (compact)'' iff ``For every set \(X\), \([0,1]^X\) is Loeb (compact)'' iff ``Every product of finite discrete spaces is Loeb (compact)''. NEWLINENEWLINE{4)} ``Tychonoff products of Loeb Hausdorff spaces are Loeb'' implies the axiom of choice for families of non-empty well-orderable sets and ``The union of a well-ordered family of well-orderable sets is well-orderable''. NEWLINENEWLINE{5)} It is relatively consistent with ZFA set theory (Zermelo-Fraenkel set theory with the axiom of extensionality weakened to allow the existence of atoms) that there exist a set \(X\) such that \(2^X\) is compact, whereas \(2^X\) is not Loeb.