Anti-Urysohn spaces

Abstract

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 $\bullet$ 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; $\bullet$ there is a locally countable AU space of size $κ$ iff $ω\le κ\le 2^{\mathfrak c}$. 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 $\bullet$ if $X$ is any SAU space then $ \mathfrak s\le |X|\le 2^{2^{\mathfrak c}}$; $\bullet$ if $\mathfrak r=\mathfrak c$ then there is a separable, crowded, locally countable, SAU space of cardinality $\mathfrak c$; \item if $λ> ω$ Cohen reals are added to any ground model then in the extension there are SAU spaces of size $κ$ for all $κ\in [ω_1,λ]$; $\bullet$ if GCH holds and $κ\leλ$ are uncountable regular cardinals then in some CCC generic extension we have $\mathfrak s=κ$, $\,\mathfrak c=λ$, and for every cardinal $μ\in [\mathfrak s, \mathfrak c]$ there is an SAU space of cardinality $μ$. The questions if SAU spaces exist in ZFC or if SAU spaces of cardinality $> \mathfrak c$ can exist remain open.