Čech-Stone compactifications of discrete spaces in ZF and some weak forms of the Boolean prime ideal theorem (Q2850623)

Let \(X\) be a non-empty set equipped with the discrete topology. In the notation and terminology of the paper under review, {\parindent=6mm \begin{itemize}\item[--] \(S(X)\) denotes the \textit{Stone space} of the Boolean algebra \(\mathcal P(X)\) (the power set of \(X\)), i.e., the set of all ultrafilters on \(X\) along with the topology having as a base the collection of all sets of the form \([A]=\{\mathcal F\in S(X):A\in\mathcal F\}\), \(A\in\mathcal P(X)\). \item[--] Let \(\delta\) be the diagonal map of the family \(\Delta_X\) of characteristic functions on \(X\), i.e., \(\delta:X\to 2^{\Delta_X}\), \(\delta(x)=(\chi_A(x))_{A\in\mathcal P(X)}\), \(x\in X\), and let \(\gamma(X)\) be the closure \(\overline{\delta(X)}\) of \(\delta(X)\) in the Tychonoff product \(2^{\Delta_X}\), where \(2=\{0,1\}\) has the discrete topology. \(\gamma(X)\) is called the \textit{Stone extension} of \(X\). It is not hard to verify that the points in \(\gamma(X)\) code the ultrafilters on \(X\). In particular, for every \(x\in X\), \(\delta(x)\) codes the principal ultrafilter on \(X\) corresponding to \(\{x\}\) (since for every \(A\in\mathcal P(X)\), \(\delta(x)(\chi_A)=1\) if and only if \(x\in A\)), and for every \(y\in\gamma(X)\setminus\delta(X)\), \(y\) codes a free (i.e., non-principal) ultrafilter on \(X\) (where \(X\) is an infinite set), specifically \(\{A\in\mathcal P(X):y(\chi_A)=1\}\) is a free ultrafilter on \(X\). The latter fact is established in Proposition 12 of the paper. \item[--] Let \(e\) be the evaluation map induced by the collection \(C\) of all functions from \(X\) to the subspace \([0,1]\) of the real line \(\mathbb R\), i.e., \(e:X\to [0,1]^C\), \(e(x)=(f(x))_{f\in C}\), \(x\in X\), and let \(\beta(X)\) be the closure \(\overline{e(X)}\) of \(e(X)\) in the Tychonoff product \([0,1]^C\). \(\beta(X)\) is called the \textit{Čech-Stone extension} of \(X\). NEWLINENEWLINE\end{itemize}} It is a well-known ZFC (Zermelo-Fraenkel set theory with the Axiom of Choice (AC)) result that for an infinite set \(X\) with the discrete topology, the spaces \(S(X)\) and \(\beta(X)\) are homeomorphic and that \(S(X)\) (hence \(\beta(X)\)) is compact. However, it is relatively consistent with ZF (Zermelo-Fraenkel set theory minus AC) that there exists an infinite set \(X\) with the discrete topology such that \(S(X)\) is \textit{not} compact (for example, \textit{A. Blass} in his paper [Bull. Acad. Pol. Sci., Ser. Sci. Math. Astron. Phys. 25, 329--331 (1977; Zbl 0365.02054)], has constructed a ZF model \(N\) in which all ultrafilters are principal, thus, for every infinite set \(X\in N\) with the discrete topology, \(S(X)\) (being homeomorphic with \(X\)) is not compact in \(N\)).NEWLINENEWLINEIn the paper under review, the authors establish that, in ZF, for any infinite set \(X\) with the discrete topology, \(S(X)\simeq\beta(X)\simeq\gamma(X)\), i.e., the latter three spaces are mutually homeomorphic (Theorems 14 and 15). The fact that the proof is carried out without invoking any choice principle makes the latter result notable.NEWLINENEWLINEBased on the above results, the authors also infer (in Corollary 16) that for an infinite set \(X\), \(\mathrm{BPI}(X)\) (every filter on \(X\) can be extended to an ultrafilter on \(X\)) is equivalent to each of the statements ``\(S(X)\) is compact'', ``\(\beta(X)\) is compact'' and ``\(\gamma(X)\) is compact'' and also that BPI (the Boolean Prime Ideal Theorem, i.e., every non-trivial Boolean algebra has a prime ideal, which is known to be equivalent to ``\((\forall X)\;\mathrm{BPI}(X)\)'') is equivalent to ``for every infinite set \(X\), the Tychonoff product \(2^{\mathcal P(X)}\) is compact''.NEWLINENEWLINEIn their paper, the authors also prove that (for an infinite set \(X\)) {\parindent=6mm \begin{itemize}\item[(1)] in \(\mathrm{ZF}+\mathrm{Id}(X)\) (where \(\mathrm{Id}(X)\) is ``\(X\) has an independent family of size \(|\mathcal P(X)|\)''; the latter is equivalent to ``the Tychonoff product \(2^{\mathcal{P}(X)}\) has a dense subset of size \(|X|\)'' -- as the authors mention to have established in their manuscript [``Independent families and some notions of finiteness'', \url{http://www.samos.aegean.gr/math/kker/papers/IndepFam-SomeFiniteness.pdf}] -- and the proof of the above fact is implicitly given in the proof of Theorem 18 of the paper under review), \(\mathrm{BPI}(X)\) is equivalent to ``\(2^{\mathcal P(X)}\) is compact'' (Theorem 18(i) and we note that (according to the results of the paper) only the implication ``\(\mathrm{BPI}(X)\) \(\to\) (\(2^{\mathcal P(X)}\) is compact)'' needs the assumption \(\mathrm{Id}(X)\) for its proof), \item[(2)] ``\(2^{\mathcal P(X)}\) is a continuous image of \(S(X)\)'' implies \(\mathrm{Id}(X)\). Consequently, under \(\mathrm{BPI}\), ``for every infinite set \(X\), \(\mathrm{Id}(X)\)'' is equivalent to ``\(\forall X\), \(2^{\mathcal P(X)}\) is a continuous image of \(S(X)\)'' (Theorem 18(ii)). NEWLINENEWLINE\end{itemize}} The paper closes with the result that it is relatively consistent with ZF that there exists an infinite set \(X\) such that \(\mathrm{BPI}(X)\) is true, whereas \(2^{\mathcal P(X)}\) is not compact (Theorem 20), thus ``for every infinite set \(X\), \(\mathrm{BPI}(X)\) \(\rightarrow\) (\(2^{\mathcal P(X)}\) is compact)'' is not a theorem of ZF and so the assumption of \(\mathrm{Id}(X)\) -- in the result of item (1) above -- cannot be dropped. To achieve their goal, the authors work in a ZF model by John Truss from his paper [\textit{J. K. Truss}, Ann. Pure Appl. Logic 73, No. 2, 191--233 (1995; Zbl 0827.03030)], in which there is an unbounded amorphous set \(X\) (i.e., \(X\) is infinite and is not the disjoint union of two infinite sets and for every positive integer \(n\), there is a partition \(\Pi\) of \(X\) into finite sets such that infinitely many members of \(\Pi\) have size greater than \(n\)); this particular infinite set \(X\) is used by the authors in order to establish their independence result.