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 ${\kappa}$ there is a space of size ${\kappa}$ in which fewer than $cf({\kappa})$ many non-empty regular closed sets always intersect; $\bullet$ there is a locally countable AU space of size $\kappa$ iff $\omega \le \kappa \le 2^{\mathfrak c}$. 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 $\lambda > \omega$ Cohen reals are added to any ground model then in the extension there are SAU spaces of size $\kappa$ for all $\kappa \in [\omega_1,\lambda]$; $\bullet$ if GCH holds and $\kappa \le\lambda$ are uncountable regular cardinals then in some CCC generic extension we have $\mathfrak s={\kappa}$, $\,\mathfrak c={\lambda}$, and for every cardinal ${\mu}\in [\mathfrak s, \mathfrak c]$ there is an SAU space of cardinality ${\mu}$. The questions if SAU spaces exist in ZFC or if SAU spaces of cardinality $> \mathfrak c$ can exist remain open.