Is the sets in density topology Euclidean $G_\delta$?

Question

It has been shown that every Borel subset of density topology X is d-$G_\delta$. I'm curious about its connection to the euclidean topology. For example, is the close/open set in the density topology a $G_\delta$ set in Euclidean topology? It seems false to me; for example, pick a non-Borel set S of Lebesgue measure 0, then it's closed in the density topology but definitely not a $G_\delta$ set in Euclidean topology. But I would like to know more about the connections between sets in density topology and its euclidean counterpart. Is there any related theorem on this subject?

Answer 1

The last chapter of Oxtoby's Measure and Category discusses category measure spaces and shortly discusses the density topology and proves the latter is regular. It is not normal (a reference for this is given). So earlier results in that chapter then imply that a countable union of nowhere dense sets (in the density topology) is still nowhere dense (and in fact being nowhere dense in this topology is equivalen to having Lebesgue measure $0$). So a nowhere dense Cantor set (a set homeomorphic to $2^{\Bbb N}$) with positive measure (which exist in $\Bbb R$) is Euclidean closed and has empty interior in the usual topology but in the density topology has non-empty interior. It's well known that the Euclidean topology is coarser (smaller) than the density topology, so this shows it's quite a bit smaller. The density topology is a pretty unusual topology, very non-metrisable.

These papers might help you to gain more (set-theoretic) topological understanding of it. This entry and its references are more about the analysis side of things.