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There are presumably three ways of characterizing the abstract Euclidean space $E^n$ that are quite different in spirit:

  1. axiomatically (with axioms concerning dimension)
  2. by the abstract Euclidean group $E(n)$ (as its symmetry group, determining $E^n$ uniquely)
  3. by presupposing a metric and requiring that the space is a maximal one with respect to the property that the $(n+1)$-dimensional Cayley-Menger determinant vanishes for all $(n+2)$-tuples of points and does not vanish for all $k$-tuples of points "in general position" for $k < n+2$.

[Having tried to "rescue" 3 by adding conditions in italics, due to Robin's comment.]

Question 1: Is it correct, actually, that $E^n$ is uniquely determined via 2 and 3?

Question 2: What are other ways of characterizing $E^n$ that are quite different in spirit?

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The hypothesis in 3 holds for all subsets of $n$-dimensional Euclidean space, so it is inadequate by itself...... –  Robin Chapman Nov 30 '10 at 14:38
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