I am thinking if I could help with my current problem. Now I have a parameterized rational function

sedeln5w

sedeln5w

Answered question

2022-06-21

I am thinking if I could help with my current problem. Now I have a parameterized rational function G ( p , z ), where p 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 R n to G , where G is a set of rational functions with indeterminate z. Then I define that a property holds on a metric space ( G , d ) if it holds on an open dense subset of G .
However, I am wondering what conditions I should put on a parameter set Θ R n , such that { G ( p , z ) | p Θ } becomes an open subset of G . Is making Θ an open dense subset of R n sufficient?

Answer & Explanation

Eli Shaffer

Eli Shaffer

Beginner2022-06-22Added 16 answers

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 R to a square with a dense image which is a countable union of segments. Even if G = G ( R n ) then it can happen that the image of Θ is countable, as for the Cantor staircase function.

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