Consider the metric space ℝ with the absolute value metric, d(x,y)=|x-y|. I need to prove whether transcendental numbers are open, closed, or neither.
I'm stuck on how to approach this. Since there is not much information on transcendental numbers, I thought maybe I can use the algebraic numbers. So, if $S$ is the set of all transcendental numbers, I consider $S^c$ which is the complement of $S$, i.e. the set of all algebraic numbers. Now I consider a polynomial $p(x)=\sum_{i=0}^na_ix^i$ Then the algebraic numbers will be {x$\in$ℝ|p(x)=0}. Is this a correct line of reasoning? Any hints in the right direction?
Thank you.