Given a covering map (continuous, surjective, closed, irreducible, compact) $f:X \to Y$, and a dense subset $D$ of $Y$, is the preimage of $D$ by $f$ a dense subset of $X$?
Edit: The question is missing context, but not details. Quite frankly, it was just a minor question pertaining to some research I'm doing — and I thought I would give StackExchange a try. My original thoughts were about the absolute of a Hausdorff space (X being the Iliadis absolute of Y), but I thought phrasing the question more generally would be better. The question is true, btw. Thanks.
Best Answer
If $D$ is dense in $Y$, then $f^{-1}(D)$ will be dense in $X$. The only property which is really used here is that $f$ is an open map. Take an open set $U$ of $X$, then its image $f(U)$ is open and thus intersects $D$. There is thus point $x\in U$ such that $f(x)\in D$, and that means $x\in U\cap f^{-1}(D)$.