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## Is normality of a Hausdorff space consequence of some property of open domains?

Let $\mathrm{r}\mathscr{O}$ be the family of open domains (regular open sets) of a topological space $\langle X,\mathscr{O}\rangle$, that is: $$A\in\mathrm{r}\mathscr{O}\iff A=\mathrm{int(\mathrm{cl(A)})}.$$

A space $\langle X,\mathscr{O}\rangle$ is normal iff for any disjoint closed sets $A$ and $B$ there are $O_1,O_2\in\mathscr{O}$ such that: $$A\subseteq O_1\quad\text{and}\quad B\subseteq O_2 \quad\text{and}\quad O_1\cap O_2=\emptyset.$$

Define: $$\tag{df \Subset} A\Subset B\iff \mathrm{cl}(A)\subseteq B,$$ and consider the following property: $$\tag{\dagger} (\forall{A,B\in \mathrm{r}\mathscr{O}})\bigl(A\Subset B\Rightarrow(\exists{C\in \mathrm{r}\mathscr{O}}) (A\Subset C\Subset B)\bigr)$$ It is easy to prove that if $\langle X,\mathscr{O}\rangle$ is normal, then it satisfies ($\dagger$). What I am interested in is whether the following is true:

If $\langle X,\mathscr{O}\rangle$ is a Hausdorff space ($T_2$-space) which satisfies ($\dagger$), then it is normal.

EDIT: Ramiro de la Vega pointed to a very nice counterexample. I have one more question: what if we require that $\langle X,\mathscr{O}\rangle$ is semiregular, that is the regular open sets form a basis for the topology? Thus what I am now asking is whether the following (weaker) statement is true:

If $\langle X,\mathscr{O}\rangle$ is a semiregular Hausdorff space which satisfies ($\dagger$), then it is normal.

EDIT: The answer to the question above is negative as well. A counterexample is relatively prime integer topology, L.A. Steen, J.A. Seebach, Jr. Counterexamples in topology, number 60.

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No.

For example let $X=\{(x,y): x,y \in \mathbb{Q}, y \geq 0 \}$ with the irrational slope topology. The clousure of any two non-empty open subsets of $X$ have non-empty intersection. This implies that $X$ satisfies ($\dagger$) trivially but it also implies that $X$ is not even $T_{2\frac{1}{2}}$ (so it is not $T_4$).

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I must say, this is a spectacularly ingenious construction in the book by Steen and Seebach; I wonder who first thought of it? However, the description given on the linked page is slightly garbled; where they say "on the $X$", it should say "on the intersection of $X$ with the $x$-axis" or words to that effect. Great example! – Todd Trimble Aug 30 at 17:37
@Todd: R. H. Bing, A connected countable Hausdorff space, Proc. Amer. Math. Soc. 4 (1953), 474, available here: dx.doi.org/10.1090/S0002-9939-1953-0060806-9 – Theo Buehler Aug 30 at 17:51
Wow, thanks Theo! – Todd Trimble Aug 30 at 18:07
@Ramiro: Thanks a lot! The counterexample is beautiful indeed. I have one more question which I included below my original one. – R.G. Aug 30 at 21:35
Relatively prime integer topology (Steen and Seebach, 60) is Hausdorff and semiregular but closures of any disjoint open sets have non-empty intersection. So the answer to the expanded version of my question is negative as well. – R.G. Sep 3 at 22:59