# gn.general topology – Does every open set contain a dense \$F_{sigma}\$ subset?

Let $$U$$ be a regular open set in a Tychonoff space $$X$$ (regular means that it is an interior of a closed set).

( In my specific situation $$U$$ is of the form $$int f^{-1}(0)$$, where $$f$$ is a continuous real-valued function on $$X$$, and $$X$$ is a Baire space (a sequence of dense open sets has a dense intersection), but I am not sure if it helps. )

Is there a sequence $${A_n}_{ninmathbb{N}}$$ of closed (in $$X$$) subsets of $$U$$ such that $$bigcup_{ninmathbb{N}} A_n$$ is dense in $$U$$?

Of course, this is the case if $$X$$ is perfectly normal (which is equivalent to every open set being $$F_{sigma}$$), or separable, but I hope a less restrictive assumption will suffice, e.g. normality.