Abstract
In set theory without the Axiom of Choice (AC), we investigate the set-theoretic strength (in terms of weak choice principles) of the following statements: “every infinite Hausdorff space has a denumerable cellular family”, “every infinite Hausdorff space has a denumerable discrete subset”, “every denumerable compact Hausdorff space has an infinite cellular family”, and “every denumerable compact Hausdorff space has an infinite discrete subset”, and answer open questions by E. Tachtsis “Infinite Hausdorff spaces may lack cellular families or discrete subsets of cardinality ℵ0” and an open question by A. Miller “Some interesting problems”, which also appears in the former paper.
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