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I have added a picture with the complete exercise. I'm interested only in c), I think I have proved a) and b).

In c) we are given a topological space $X_0$ which is semi-regular (i.e. there exists a base for the topology whose open set satisfy the relation $\mathring{\overline{U}}=U$) and another topological space $X$ (the underlying set is always the same). Then, we have to prove that $X^*=X_0$ if and only if there exists a family $\mathfrak M$ of dense subsets of $X$ such that every finite intersection of sets of $\mathfrak M$ belongs to $\mathfrak M$ and such that the topology on $X$ is generated by the union of $\mathfrak M$ and the open sets of $X_0$ (here $X^*$ is the topology whose base are the regular open sets in $X$). Hence, we have to manage to construct examples of topologies finer than regular Hausdorff spaces which are not regular.

To prove the result, the exercise gives us a hint: We should consider the dense open subsets of $X$ and notice that every open set in $X$ can be written as the intersection of a dense open set in $X$ with an open set in $X_0$.

I'm not sure how to start with. I know that, if $D$ is a dense subset, then $\overline U = \overline{U\cap D}$, but I don't think this helps at all.

Any hint will be grateful. Thanks.

EDIT:

The axiom $\mathrm{O_{III}}$ is the condition of regularity: For each closed set $F$ and each point $x\in X\setminus F$, there are disjoint open sets containing $x$ and $F$, respectively.

CONTEXT:

My goal is th goal of the exercise: I want to give finer topologies than regular Hausdorff that aren't regular Hausdorff. I think it is interesting because finer topologies than $T_0$, $T_1$, $T_2$, $T_{21/2}$ or completely Hausdorff are $T_0$, $T_1\dots$, resp. But it doesn't happen for $T_3$, and I would like to know why. My guess is that making finer the topology may appear new closed sets that doesn't verify the $\mathrm{O_{III}}$ axiom; namely, we have made the topology finer, but not enough, so we have create new closed sets but not enough open sets to separatd them from points.

Here is the complete exercise

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    $\begingroup$ Your first sentence has a wrong definition for a semi regular space. $\endgroup$ – Henno Brandsma Mar 7 at 5:02
  • $\begingroup$ What is Bourbaki's Axiom O$_{III}$? $\endgroup$ – DanielWainfleet Mar 7 at 6:54
  • $\begingroup$ @HennoBrandsma Yes, that's true. Only the sets of a certain base for the topology of $X_0$ satisfy that. Thanks. I hope this doesn't matter for my previous work. I guess no but... $\endgroup$ – Dog_69 Mar 7 at 6:58
  • $\begingroup$ @DanielWainfleet It is regularity: For each closed set $F$ and each $x\in X\setminus F$ there exists disjoint open sets containing $F$ and $x$. $\endgroup$ – Dog_69 Mar 7 at 7:00
  • $\begingroup$ The common modern notation for regular space is $T_3$ space. $\endgroup$ – DanielWainfleet Mar 7 at 7:31

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