Large strongly anti-Urysohn spaces exist

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