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Here a proper metric space is a metric space such that all closed balls are compact. My question is are all Gromov hyperbolic spaces proper metric spaces? I only know the rudimentary definitions of these things which is why I am asking, I didn't see it referenced anywhere and I was curious.

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No. For instance, a hedgehog space with infinitely many spines is Gromov hyperbolic but is not proper (for instance, any closed ball around the center point contains an infinite closed discrete subspace by picking one point in each spine, each the same distance from the center point).

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Even simpler, any infinite set $X$ endowed with the discrete metric, ie. $$(x,y) \mapsto \left\{ \begin{array}{l} 0 & \text{if $x=y$} \\ 1 & \text{otherwise} \end{array} \right.,$$ provides an example of a non proper Gromov hyperbolic space.

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