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Given a (nonempty) set $X$ and a subset $E \subseteq X$, is it possible to find two non-equivalent metrics $d_1, d_2$ such that $E$ is dense in $(X,d_1)$, but $E$ is not dense in $(X,d_2)$?

Remark: If I am not wrong, the requirement that $d_1,d_2$ are not equivalent is necessary.

Thanks for any comments!

user237522
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Yes. Hint: There is a simple metric in which no proper subset can be dense, because all points are open.

Elle Najt
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    You mean the discrete topology? – user237522 Apr 22 '17 at 23:32
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    @user237522 Yes. – Elle Najt Apr 22 '17 at 23:33
  • I must confess that I had in mind $X=k[x,y]$, $E=k[p,q]$, where the Jacobian of ${p,q}$ is in $k^*$. Each of $X$ and $E$ is dense in $Y=k[[x,y]]$, so in the induced topology, $k[p,q] \subseteq k[x,y]$ is dense. But if I do not like to have infinite sums, then I have not yet found a metric for $k[x,y]$ such that $k[p,q]$ is dense in $k[x,y]$; I suspect that this is impossible, since, if I am not wrong, all metrics are equivalent. See also https://en.wikipedia.org/wiki/Formal_power_series – user237522 Apr 22 '17 at 23:44
  • @user237522 Then you should ask that question. What is the Jacobian of ${p, q}$? What is the relationship of $p$ and $q$ to $x$ and $y$? Are $p$ $q$ polynomials? – Elle Najt Apr 22 '17 at 23:49
  • Yes, $p,q \in k[x,y]$ are such that $p_xq_y-p_yq_x \in k^*$ (partial derivatives). – user237522 Apr 22 '17 at 23:52
  • I wonder why the OP specified "a metrizable space $(X,\mathcal T)$" and not simply "a set $X$", if he was interested in arbitrary metrics, not just metrics inducing the topology $\mathcal T$? (Of course, if the metrics have to be compatible with $\mathcal T,$ then they must have the same dense sets, since denseness is topological.) – bof Apr 23 '17 at 00:48
  • @bof, thanks! perhaps I should change my question according to your comment. Please, any remarks about my above particular case $k[p,q] \subseteq k[x,y]$? see https://math.stackexchange.com/questions/2247342/can-kp-q-be-dense-in-kx-y-where-p-xq-y-p-yq-x-in-k – user237522 Apr 23 '17 at 01:19