Extending compact topologies to compact Hausdorff topologies in \(\mathbf {ZF}\) (Q645188)

The core of the research in this interesting paper concerns the following question posed by Murray Bell (see also \textit{A. W. Miller}'s list of problems on the Axiom of Choice (AC) on his website [``Some interesting problems'', \url{http://www.math.wisc.edu/~miller/res/problem.pdf}], Section 13, ``not AC'', p. 16, Problem 13.1): Does the statement ``For every family \(\{A_i:i\in I\}\) of non-empty sets there exists a family \(\{T_i:i\in I\}\) such that, for every \(i\in I\), \((A_i,T_i)\) is a compact \(T_2\) space'' imply AC? (The authors denote that statement by (A).) In case (A) does not imply AC, what principles of choice is (A) equivalent to? It is known that the conjunction of (A) with BPI (the Boolean Prime Ideal theorem, i.e., every Boolean algebra has a prime ideal) is, in ZF (the Zermelo-Fraenkel set theory minus AC), equivalent to AC (we recall here that BPI is equivalent to the statement ``The Tychonoff product of compact \(T_2\) spaces is compact'', a fact which was was proved by Łos/Ryll-Nardzewski as well as by Rubin/Scott), hence (A) is not provable in ZF since it is a well-known result by Halpern and Lévy that BPI is strictly weaker than AC in ZF. Investigating the deductive strength of (A) and of related statements, the authors prove, among other results, the following within ZF: (A) implies there are no amorphous sets (a set \(X\) is called amorphous if \(X\) is infinite and cannot be written as the disjoint union of two infinite sets). AC is equivalent to the statement ``For every family \(\{A_i:i\in I\}\) of non-empty sets there exists a family \(\{T_i:i\in I\}\) such that, for every \(i\in I\), \((A_i,T_i)\) is a compact, scattered \(T_2\) space'' (a space \((X,T)\) is scattered if, for each non-empty subspace \((Y,T_Y)\) of \(X\), \(\{y\in Y:y\text{ is isolated in }(Y,T_Y)\}\neq\emptyset\)). AC is equivalent to the conjunction of the statements ``For every set \(X\), every compact \(R_1\) topology (its \(T_0\)-reflection is \(T_2\)) on \(X\) can be extended to a compact \(T_2\) topology'' and ``There exists an infinite set \(X\) and a non-principal ultrafilter on \(X\)''. (We recall here that it is known that it is not true in general (even in \(\text{ZFC} = \text{ZF}+\text{AC}\)) that every compact \(T_1\) topology on a set \(X\) can be extended to a compact \(T_2\) topology; the authors also supply a counterexample in Example 1 on top of p. 2281 of their paper.)