User:IssaRice/Isometry in metric spaces

when playing around with metric spaces, one might notice that certain metric spaces can be "modeled" by other metric spaces. For instance, let $X = {1, 2, 3}$ be a set, and let $(X, d_{d i s c})$ be the discrete metric on X. Then we can "model" this metric space by the familiar Euclidean metric on ${(0, 0), (1, 0), (1 / 2, \sqrt{3} / 2)}$ (the set looks like an equilateral triangle with edge length 1). Similarly, with $Y = {1, 2, 3, 4}$ , the metric space $(Y, d_{d i s c})$ can be modeled by ${(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)}$ with the sup norm metric, by ${(1 / 2, 0, 0, 0), (0, 1 / 2, 0, 0), (0, 0, 1 / 2, 0), (0, 0, 0, 1 / 2)}$ with the taxicab metric, or by ${(\sqrt{1 / 2}, 0, 0, 0), (0, \sqrt{1 / 2}, 0, 0), (0, 0, \sqrt{1 / 2}, 0), (0, 0, 0, \sqrt{1 / 2})}$ with the Euclidean metric.

What, precisely, do we mean by "modeling" metric spaces? Let $(X, d_{X})$ be a metric space, and let $(Y, d_{Y})$ be a metric space. Then it seems like we want to say that given points $x, x^{'} \in X$ , we have $d_{X} (x, x^{'}) = d_{Y} (y, y^{'})$ , where $y, y^{'}$ are the corresponding points in $Y$ . We want some bijection f that maps these points for us. So our final notion is this: Y models X iff there exists some bijection $f : X \to Y$ such that $d_{X} (x, x^{'}) = d_{Y} (f (x), f (x^{'}))$ for all $x, x^{'} \in X$ .

And the above is just the definition of isometric metric spaces.

Questions:

is the discrete metric always isometric to R^something with the taxicab/sup norm/Euclidean metric? Given $X$ , we can consider $Y = {f ∣ f : X \to R}$ and define $d (f, g) = \sum_{x \in X} | f (x) - g (x) |$ or something, where $x \mapsto f$ such that $f (x) = 1 / 2$ and zero everywhere else. (and similarly for the other metrics)
call X more powerful than Y if X can model more metric spaces by taking appropriate subspaces of itself. are the euclidean metric, taxicab metric, and sup norm metric equally powerful?
is there a metric space that cannot be modeled by the Euclidean metric?