Anti-Urysohn spaces
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 κ there is an AU space of cardinality κ, 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 2c 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 2c.