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) ery family ( A i ) i ∈ I of sets there exists a family ( T i ) i ∈ I such that for every i ∈ I ( A i , T i ) is a compact T 2 space. ery family ( A i ) i ∈ I of sets there exists a family ( T i ) i ∈ I such that for every i ∈ I ( A i , T i ) is a compact, scattered, T 2 space. ery set X, every compact R 1 topology (its T 0 -reflection is T 2 ) on X can be enlarged to a compact T 2 topology. w:(a) plies every infinite set can be split into two infinite sets. f AC. d “there exists a free ultrafilter” iff AC. so show that if the topology of certain compact T 1 spaces can be enlarged to a compact T 2 topology then (1) holds true. But in general, compact T 1 topologies do not extend to compact T 2 ones.