What is the most general definition of a coordinate system? What is the most general definition of a coordinate system?    
Specificly:


*

*given a suitably general metric space $(\mathcal S, s)$ consisting of a set $\mathcal S$ of elements
(for instance: a set of events) together with a function $s : \mathcal S \times \mathcal S \rightarrow \mathbb R$, and

*given a function $\varphi : \mathcal S \rightarrow \mathbb R^n$, for a natural number $n \ge 1$,
such that there exists the inverse function $\varphi^{-1}$
(namely any such function $\varphi$, without any specific further conditions),
do $\mathcal S$, $s$, and $\varphi$ together constitute a coordinate system?
Or which additional conditions would have to be imposed on function $\varphi$, to this end? 
 A: In principle, yes, $\varphi$ is a coordinate system, in that it tags every member of $\mathcal S$ with an appropriate $n$-tuple of real numbers. If you know a set of coordinates $(c_1,\ldots,c_n)$, you can use $\varphi^{-1}$ to find the corresponding point in $\mathcal S$.
In practice, this will not be useful at all, mostly because the amount of real numbers you can actually "know" is very, very limited. The situation you have described includes all invertible functions $\varphi:\mathbb R\to\mathbb R$, and this is a very big, very scary set. (To see the sort of stuff you allow by doing this, grab a copy of Counterexamples in analysis, in print or online.)
To provide a minimally useful coordinate system, we usually require at the very least that $\varphi$ and its inverse be continuous. This permits one to permit finite knowledge about a point with only finite knowledge about its coordinates, and vice versa, which is the very minimum necessary to perform anything that looks like an experiment.
In the end, though, this is strictly a semantics issue. Different authors will mean different things by the term, which is why we prefer to use unambiguous terms like homeomorphism and diffeomorphism. Authors which use the term coordinate system in any rigorous environment will define what they mean by the term; if they don't, it is assumed to be as well-behaved as necessary for the purpose at hand.
There are in the end no hard answers to your question. "Coordinate system" means what you want it to mean. However, you do need to be aware that this is distinct from the more subtle notion of "useful coordinate system", which depends on what you're trying to do; provide too little structure for your purpose and you will fail. Ask yourself "what do I want this coordinate system for?" and that will tell you what other conditions you need to impose on it.
