Is normality of a Hausdorff space consequence of some property of open domains? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T19:58:01Z http://mathoverflow.net/feeds/question/105938 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/105938/is-normality-of-a-hausdorff-space-consequence-of-some-property-of-open-domains Is normality of a Hausdorff space consequence of some property of open domains? R.G. 2012-08-30T12:40:48Z 2012-09-03T22:54:39Z <p>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: $$A\in\mathrm{r}\mathscr{O}\iff A=\mathrm{int(\mathrm{cl(A)})}.$$</p> <p>A space $\langle X,\mathscr{O}\rangle$ is <i>normal</i> iff for any disjoint closed sets $A$ and $B$ there are $O_1,O_2\in\mathscr{O}$ such that: $$A\subseteq O_1\quad\text{and}\quad B\subseteq O_2 \quad\text{and}\quad O_1\cap O_2=\emptyset.$$</p> <p>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:</p> <blockquote> If $\langle X,\mathscr{O}\rangle$ is a Hausdorff space ($T_2$-space) which satisfies ($\dagger$), then it is normal. </blockquote> <p>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:</p> <blockquote> If $\langle X,\mathscr{O}\rangle$ is a semiregular Hausdorff space which satisfies ($\dagger$), then it is normal. </blockquote> <p>EDIT: The answer to the question above is negative as well. A counterexample is <i>relatively prime integer topology</i>, L.A. Steen, J.A. Seebach, Jr. <i>Counterexamples in topology</i>, number 60.</p> http://mathoverflow.net/questions/105938/is-normality-of-a-hausdorff-space-consequence-of-some-property-of-open-domains/105956#105956 Answer by Ramiro de la Vega for Is normality of a Hausdorff space consequence of some property of open domains? Ramiro de la Vega 2012-08-30T15:36:03Z 2012-08-30T15:36:03Z <p>No. </p> <p>For example let $X=\{(x,y): x,y \in \mathbb{Q}, y \geq 0 \}$ with the <a href="http://brubeck.jdabbs.com/spaces/irrational-slope-topology/" rel="nofollow">irrational slope topology</a>. 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 $T_4$).</p>