Is every regular separable space normal? Here are some (standard, I think) definitions:
- A space is called regular if given any nonempty closed set $F$ and any point $x$ that does not belong to $F$, there exists a neighbourhood $U$ of $x$ and a neighbourhood $V$ of $F$ that are disjoint.
- A space is called normal if, given any disjoint closed sets $E$ and $F$, there are open neighbourhoods $U$ of $E$ and $V$ of $F$ that are also disjoint.
- A space is called separable if it contains a countable, dense subset.