Is the intersection of a regular open with a dense subspace regular open in the subspace? Let $X$ be a topological space (no assumptions about separation), $U$ be a regular open subset of $X$ (i.e. $\mbox{int}\,\mbox{cl}\, U = U$) and $D$ a dense subset of $X$. Is $D \cap U$ a regular open of $D$, i.e. is $\mbox{int}_D \mbox{cl}_D (D \cap U) = D \cap U$?
 A: I do not know if in some point I make some mistake. $U\cap D$ should be open in the induced topology over $D$ (this is due to the density of $D$ inside $X$). 
Suppose now that $\exists x\in\mbox{int}_D\mbox{cl}_D (U\cap D) \setminus (U\cap D)  $.
This means:


*

*$x\notin U$

*$x\in \mbox{cl}_D(U\cap D) $

*$\exists \bar{U}^{(D)}_x$ open neighbourhood of $x$ in $D$ such that $ \bar{U}^{(D)}_x\subseteq \mbox{cl}_D(U\cap D)$


From 2 we have that $\forall U^{(D)}_x$, $U^{(D)}_x\cap D\cap U \neq \emptyset$ and in particular we have $U^{(D)}_x\cap U\neq \emptyset$. Since $D$ is dense it follow that $\forall U^{(X)}_x$ open in $X$ we have $U^{(X)}_x\cap U \neq \emptyset$.
Since $U$ is regular in $X$ and since $x$ is a point in the frontier of $U$, it follows that $\nexists U^{(X)}_x$ such that $U^{(X)}_x\subseteq \mbox{cl}_X(U)$ and you can see that this implies   $\nexists\  U^{(X)}_x\cap D$ (or $U^{(D)}_x$) such that $U^{(D)}_x\subseteq \mbox{cl}_D(U)$. This contradicts point 3.
This final statement follows from two facts. 
First we have that $\mbox{cl}_D(U)\subseteq \mbox{cl}_X(U)$. Indeed assume by contradiction that $\exists x \in \mbox{cl}_D(U)\setminus \mbox{cl}_X(U) $ . This means $\exists U^{(X)}_x$ such that $U^{(X)}_x\cap U = \emptyset$ and in particular $(U^{(X)}_x\cap D)\cap U= \emptyset$. If we set $U^{(D)}_x=(U^{(X)}_x\cap D)$, we obtain a contradiction because $(D\cap U^{(D)}_x)=\emptyset$.
Second we have that  saying  $\nexists U^{(X)}_x$ such that $U^{(X)}_x\subseteq \mbox{cl}_X(U)$ is equal to say that  $\forall U^{(X)}_x $, $U^{(X)}_x\cap (\mbox{cl}_X(U))^c\neq\emptyset\ $ (the complementar of the closure in $X$ of $U$). Note that this last set is open, and so for density we have $\forall U^{(D)}_x $, $U^{(D)}_x\cap (\mbox{cl}_X(U))^c\neq\emptyset\ $. 
I think this should work.  
