Large strongly anti-Urysohn spaces exist

Abstract

As defined in [3], a Hausdorff space is strongly anti-Urysohn (in short: SAU) if it has at least two non-isolated points and any two infinite closed subsets of it intersect. Our main result answers the two main questions of [3] by providing a ZFC construction of a locally countable SAU space of cardinality 2c. The construction hinges on the existence of 2c weak P-points in ω⁎, a very deep result of Ken Kunen.It remains open if SAU spaces of cardinality >2c could exist, while it was shown in [3] that 22c is an upper bound. Also, we do not know if crowded SAU spaces, i.e. ones without any isolated points, exist in ZFC but we obtained the following consistency results concerning such spaces.(1)It is consistent that c is as large as you wish and there is a locally countable and crowded SAU space of cardinality c+.(2)It is consistent that both c and 2c are as large as you wish and there is a crowded SAU space of cardinality 2c.(3)For any uncountable cardinal κ the following statements are equivalent:(i)κ=cof([κ]ω,⊆).(ii)There is a locally countable and crowded SAU space of size κ in the generic extension obtained by adding κ Cohen reals.(iii)There is a locally countable and countably compact T1-space of size κ in some CCC generic extension.