Extending compact topologies to compact Hausdorff topologies in ZF

摘要

Within the framework of Zermelo-Fraenkel set theory ZF, we investigate the set-theoretical strength of the following statements: (1) For every 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. (2) For every 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. (3) For every set X, every compact R_1 topology (its To-reflection is T_2) on X can be enlarged to a compact T_2 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 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.