If $f\colon X\to Y$ is an almost one-to-one continuous map from the connected space $X$ onto $Y$, then, is it a homeomorphism?

Question

Let $X,Y$ be metrizable compact spaces, with $X$ connected, and $f\colon X \to Y$ a continuous function onto $Y$. Suppose $f$ is almost one-to-one in the following sense: there exists $Y' \subseteq Y$, a dense $G_\delta$, such that $\# f^{-1}(y) = 1$ for all $y \in Y'$. Then, can happen that $Y\setminus Y'$ is dense in $Y$?

I have this situation in another problem and it is hard for me to image a that $Y$ has both a dense set of point where $f$ has a unique preimage and a dense set of points where $f$ has two or more preimages.

Answer 1

Yes, it can. For instance, let $X=\{(x,y)\setminus [0,1]^2:y\le D_M(x)\}$, where $D_M(x)$ is a modified Dirichet function and $f$ be the projection of $X$, $(x,y)\mapsto x$.