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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.

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  • $\begingroup$ Do you have an example of a space with only the first property? $\endgroup$
    – M. Winter
    Commented 8 hours ago
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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? $\endgroup$
    – M. Winter
    Commented 8 hours ago
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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? $\endgroup$
    – Fedor Petrov
    Commented 8 hours ago
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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? $\endgroup$
    – Denis T
    Commented 7 hours ago
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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? $\endgroup$
    – Denis T
    Commented 7 hours ago

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It 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). $\endgroup$
    – Anixx
    Commented 8 hours ago

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