Extending compact topologies to compact Hausdorff topologies in ZF

Within the framework of Zermelo–Fraenkel set theory ZF, we investigate the set-theoretical strength of the following statements:(1)For every family (Ai)i∈I of sets there exists a family (Ti)i∈I such that for every i∈I (Ai,Ti) is a compact T2 space.(2)For every family (Ai)i∈I of sets there exists a family (Ti)i∈I such that for every i∈I (Ai,Ti) is a compact, scattered, T2 space.(3)For every set X, every compact R1 topology (its T0-reflection is T2) on X can be enlarged to a compact T2 topology.We show:(a)(1) implies every infinite set can be split into two infinite sets.(b)(2) iff AC.(c)(3) and “there exists a free ultrafilter” iff AC. We also show that if the topology of certain compact T1 spaces can be enlarged to a compact T2 topology then (1) holds true. But in general, compact T1 topologies do not extend to compact T2 ones.