Anti-Urysohn spaces

All spaces are assumed to be infinite Hausdorff spaces. We call a space anti-Urysohn (AU in short) iff any two non-empty regular closed sets in it intersect. We prove that•for every infinite cardinal κ there is a space of size κ   in which fewer than cf(κ)cf(κ) many non-empty regular closed sets always intersect;•there is a locally countable AU space of size κ   iff ω≤κ≤2cω≤κ≤2c.A space with at least two non-isolated points is called strongly anti-Urysohn (SAU in short) iff any two infinite closed sets in it intersect. We prove that•if X   is any SAU space then s≤|X|≤22cs≤|X|≤22c;•if r=cr=c then there is a separable, crowded, locally countable, SAU space of cardinality cc;•if λ>ωλ>ω Cohen reals are added to any ground model then in the extension there are SAU spaces of size κ   for all κ∈[ω1,λ]κ∈[ω1,λ];•if GCH holds and κ≤λκ≤λ are uncountable regular cardinals then in some CCC generic extension we have s=κs=κ, c=λ, and for every cardinal μ∈[s,c]μ∈[s,c] there is an SAU space of cardinality μ.The questions if SAU spaces exist in ZFC or if SAU spaces of cardinality >c>c can exist remain open.