From open dense parameter set to open dense set of rational functions

Question

I am thinking if I could help with my current problem. Now I have a parameterized rational function $G(p,z)$, where $p \in \mathbb{R}^n$ denotes the coefficients (parameters) of the rational function, and $z$ denotes the indeterminate of the rational function which lies in complex domain.

I regard $G(p,z)$ as a mapping from $\mathbb{R}^n$ to $\mathbb{G}$, where $\mathbb{G}$ is a set of rational functions with indeterminate $z$. Then I define that a property holds on a metric space $(\mathbb{G},d)$ if it holds on an open dense subset of $\mathbb{G}$.

However, I am wondering what conditions I should put on a parameter set $\Theta \subseteq \mathbb{R}^n$, such that $\{G(p,z)| p\in \Theta\}$ becomes an open subset of $\mathbb{G}$. Is making $\Theta$ an open dense subset of $\mathbb{R}^n$ sufficient?

Thanks

Answer 1

I guess that the conditions should be put also on the map $G$ and they should be rather strong taking into account the following examples. It is easy to construct a continuous map from $\Bbb R$ to a square with a dense image which is a countable union of segments. Even if $\Bbb G=G(\Bbb R^n)$ then it can happen that the image of $\Theta$ is countable, as for the Cantor staircase function.