Anti-Urysohn spaces

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All spaces are assumed to be infinite Hausdorff spaces. We call a space "anti-Urysohn" (AU in short) iff any two non-emty 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) many non-empty regular closed sets always intersect; there is a locally countable AU space of size κ iff ω≤ 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|≤ 2²ᶜ; if r=c then there is a separable, crowded, locally countable, SAU space of cardinality c; if λ> 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]; if GCH holds and κ≤λ are uncountable regular cardinals then in some CCC generic extension we have s=κ, c=λ, and for every cardinal 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 can exist remain open.