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  • $\begingroup$ why do you want a metric space? more specifically, your system seems to be a global coordinate system (a unique map from all the space to all $\mathbb{R}^n$; which topological/metrical properties of the latter do you want to be preserved by the inverse map? $\endgroup$ Commented Apr 28, 2015 at 19:07
  • $\begingroup$ @yuggib: "why do you want a metric space?" -- Any suitably generalized metric space. So we may speak of some sort of "system" at all. "your system seems to be a global coord. system; a unique map from all the space to all $\mathbb R^n$" -- That's not the intended meaning. I mean $\varphi$ to be a unique map from all the (non-empty) space to some (non-empty) subset of $\mathbb R^n$, or all $\mathbb R^n$. "which topological/metrical properties of the latter do you want to be preserved by the inverse map?" -- None which aren't outright required for speaking of a "coordinate system". $\endgroup$ Commented Apr 28, 2015 at 19:19
  • $\begingroup$ 1) Well, the least requirement possible would be a collection of points without any additional structure. Or you may think of a space with topology but without a metric, for example. 2) the coordinate system is global because maps all $S$ into the subset of $\mathbb{R}^n$; in manifolds usually the mapping is local (in the sense that $S$ is only locally (in a neighborood of each point) isomorphic to a subset of $\mathbb{R}^n$). 3) The requirements of a cooridnate system depend, in my opinion, to what you need them for; as I said above you may only need them to be a set (a collection of points). $\endgroup$ Commented Apr 28, 2015 at 19:38
  • $\begingroup$ @yuggib: "1 [...] the least requirement possible would be a collection of points without any additional structure." -- Presumably no "structure" in addition to $$\mathfrak g : \mathbb R^n \times \mathbb R^n \rightarrow \mathbb R, \qquad \mathfrak g[~\mathbf x_a, \mathbf x_b~] \mapsto s[~\varphi^{-1}[~\mathbf x_a~], \varphi^{-1}[~\mathbf x_b~]~].$$ "[...] 3) The requirements of a coordinate system depend, in my opinion, to what you need them for" -- Well, the larger point of my question is to establish that, in Physics, there is no genuine need for coordinates. $\endgroup$ Commented Apr 28, 2015 at 19:55
  • $\begingroup$ Even if there is no necessity of coordinates, they are quite useful; also in relativity it is postulated that the (coordinate free) space-time is locally isomorphic to the minkowski space-time (and thus local coordinates emerge quite naturally). Anyways I do not see why do you insist in putting the metric as a necessary requirement to define "coordinates"; if I define the coordinates as a subset of $\mathbb{R}^n$ set-isomorphic to my given set (or a part of it), I am defining an identification of the points of my set with the $n$-tuples of reals, that I may call coordinates... $\endgroup$ Commented Apr 28, 2015 at 20:10