I'm trying to show that for any topological space $X$ and any $ O\subseteq X$ with $O$ an open set, there is an regular open set $W$ such that $ O \subseteq W$ and $O$ is dense in $W$.
I'm not quite sure what "$O$ is dense in $W$" means. Does this typically mean that $O$ is dense in the subspace topology induced by $W$ or for any nonempty open set contained in $W$, $O$ meets it?
I appreciate any clarification
Edit: I believe that I can just let $W = $ Int(Cl($O$)) as the regular open set, but I'm still trying to show the density part.