  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.
