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Is it possible to 'find' all metrics on $\mathbb{C}[x,y]$?

If not, at least:

Is there a metric on $\mathbb{C}[x,y]$ such that the sub-algebra $\mathbb{C}[p,q]$, where $p,q \in \mathbb{C}[x,y]$ satisfy $p_xq_y-p_yq_x \in \mathbb{C}^*$, is dense in $\mathbb{C}[x,y]$?

How much this depends on metrics on $\mathbb{C}$? (if at all).

This question is relevant.

user237522
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    How do you want the metric to relate to the algebraic structure? Otherwise, the first question really does not make much sense (since $\mathbb{C}[x,y]$ without structure is just another set with the cardinality of the continuum), and there are way too many metrics. – Lukas Geyer Apr 25 '17 at 01:07
  • @LukasGeyer, thanks for your remark. So let us think of $\mathbb{C}[x,y]$ as a topological ring. Please, do you have any hints for an answer? – user237522 Apr 25 '17 at 01:18
  • @LukasGeyer, out of curiosity, can you please give an example of a metric on $\mathbb{C}[x,y]$ for which $\mathbb{C}[p,q] \subseteq \mathbb{C}[x,y]$ is dense (without assuming the continuity of addition and multiplication). Am I missing something trivial? – user237522 Apr 25 '17 at 01:31
  • I am not sure what I am missing, but for $p(x,y)=x$ and $q(x,y)=y$ you have that $\mathbb{C}[p,q]=\mathbb{C}[x,y]$, so it is certainly dense. Otherwise, if you have any infinite subset of your set, you can always find a bijection to the real line such that the image of the subset contains the rational numbers. Pulling the usual metric on the real line back to your space makes that infinite subset dense. – Lukas Geyer Apr 25 '17 at 05:29

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