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Related to a more historically inclined question I'd like to ask, if the language and axioms of Hilbert's geometry do suffice to first state and then prove that in the standard model (Euclidean space) every circle can be (continuously) collapsed to a point, i.e. that Euclidean space has genus $0$. Or is this beyond Euclidean geometry?

(If it cannot be proved there must be models of genus $\neq 0$?)

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  • $\begingroup$ I'm fairly sure I can't give a definitive answer this question, but regardless I think that some clarification is needed. What do you mean by proving Euclidean space has genus 0 in the language of Hilbert's geometry? I ask because continuity and continuous maps from the circle cross the interval to Euclidean space are definitely not notions of Hilbert's geometry. Continued below. $\endgroup$ – jgon Nov 5 '18 at 14:23
  • $\begingroup$ This is a crucial point though because geometric circles and topological circles do not coincide as notions at all. Consider the 2-torus with metric induced by the quotient $\Bbb{R}^2/\Bbb{Z}^2$. Then lines of rational slope are topologically circles, but not geometrically. Moreover, every geometric circle can be continuously contracted to a point, since geometric circles are the images of geometric circles in $\Bbb{R}^2$. However the torus of course doesn't have genus 0. (Depending on how you define circles in a Riemannian manifold I guess. I don't do much Riemannian geometry.) $\endgroup$ – jgon Nov 5 '18 at 14:24
  • $\begingroup$ @jgon: Would geometric and topological closed curves coincide? $\endgroup$ – Hans-Peter Stricker Nov 5 '18 at 14:55
  • $\begingroup$ I'm not sure. More precisely, I'm not sure how one would define a geometric closed curve in the language of Hilbert's geometry. $\endgroup$ – jgon Nov 5 '18 at 14:57
  • $\begingroup$ This may be recklessly naive, but wouldn't it be as simple as proving: Given any circle $C$ with center $O$ and radius $R$, and given any $0<r<R$, there exists a circle centered at $O$ with radius $r$? That seems to capture the idea of "a circle can be collapsed to a point" pretty naturally without needing to formally define continuity. $\endgroup$ – mweiss Nov 5 '18 at 15:15

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