On (non-Menger) spaces whose closed nowhere dense subsets are Menger

A space $X$ is od-Menger if it satisfies ${\mathsf{U}}_{\mathsf{f}\mathsf{i}\mathsf{n}}({\mathrm{\Delta }}_X,{\mathcal{O}}_X)$, where ${\mathcal{O}}_X,{\mathrm{\Delta }}_X$ are the collection of covers of $X$ by respectively open subsets and open dense subsets. We show that under CH, there is a refinement of the usual topology on a subset of the reals which yields a hereditarily Lindelöf, od-Menger, non-Menger, $0$-dimensional, first countable space. We also investigate the properties of spaces which are od-Menger but not Menger.