T3 space which does not satisfy certain condition By a $T_3$ topological space I mean a space in which a point $x$ and a closed set $A$ which does not contain $x$ can be separated by their open neighborhoods, i.e. there are $U,V\in\mathscr{O}$ such that $A\subseteq U$ and $x\in V$ and $U\cap V=\emptyset$.
Could you please point to me some classical example of a space which satisfies the following condition ($\mathscr{O}^+=\mathscr{O}\setminus\{\emptyset\}$ and $\mathrm{Cl}$ is the standard closure operator):
$$\forall_{U\in\mathscr{O}^+}\exists_{V\in\mathscr{O}^+}\,\mathrm{Cl}\, V\subseteq U$$
but is not $T_3$?

A very simple example is the one that I described in this answer: it has the desired property, because the isolated points are a dense subset. In fact any non-regular $T_1$ space with a dense set of isolated points has the property.
