An anti-Urysohn (AU) space is a Hausdorff space in which any two {\em regular} closed sets intersect

and a strongly anti-Urysohn (SAU) space is a Hausdorff space that has at least two non-isolated points and in which any two {\em infinite} closed sets intersect.

For every infinite cardinal $\kappa$ there is an AU space of cardinality $\kappa$, but if $X$ is SAU then

$s \le |X| \le 2^2^c$.

In recent joint work with Shelah, L. Soukup and Szentmiklóssy we constructed a locally countable SAU space of size $2^c$ in ZFC. This example is scattered and it is open if a crowded SAU space exists in ZFC. However, we have a consistent example of a SAU space of size $2^c$.