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Can a space with the following properties exist? Any examples?

  1. The number of geodesic paths between any two points is infinite but countable.

  2. The infimum of the geodesic path lengths between any two points is zero.

  3. For any two points there is a maximum finite geodesic path length.

  4. The maximum geodesic path length between two points has properties, similar to the distance in metric space, except it is, oviously not minimal, but maximal geodesic path.

I want to point out that this hypothetical space has properties, similar to a torus, such as Clifford torus, for example, but with minimum and infimum replaced with miximum and supremum and zero with infinity.

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  • $\begingroup$ Do you have an example of a space with only the first property? $\endgroup$ Commented Dec 18, 2024 at 11:39
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    $\begingroup$ I am not experienced in thinking about geodesic paths, but if between $x$ and $y$ there are two geodesic paths $\gamma$ and $\rho$, then aren't $\gamma(\rho^{-1}\gamma)^n$ also geodesic paths of arbitrarily large length, violating 3? $\endgroup$ Commented Dec 18, 2024 at 12:00
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    $\begingroup$ @Anixx is not the length of every path (including geodesic) between two points in a metric path always not less than the distance between them? $\endgroup$ Commented Dec 18, 2024 at 12:34
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    $\begingroup$ @Anixx So why don't you define precisely what you mean by words "metric space" and "geodesic" if you want to use them in non-classic meaning? $\endgroup$ Commented Dec 18, 2024 at 13:08
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    $\begingroup$ @Anixx The one which is present in absence of metric. Are you talking about Busemann spaces, G-spaces, something else? $\endgroup$ Commented Dec 18, 2024 at 13:13

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A non-empty metric space that satisfies all of these conditions does not exist because condition 2 will force the geodesic metric space to be a point but a metric space with a single point violates condition 1.

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  • $\begingroup$ Of course, the classic definition of distance between two points would be zero, but we are not talking about distance but about geodesic paths. We can define distance as the biggest geodesic path (rather than smallest as in classic metric spaces). $\endgroup$ Commented Dec 18, 2024 at 12:31

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