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Given a metric space, we can check what the largest number of equidistant points are (ie, such that the distance between any two of these points is the same). Of course, this might not be finite, as in the case of the discrete metric on an infinite space. This is not an invariant of the induced metric, however, as we can see by looking at the usual euclidian distance in $\mathbb{R}^2$ and the metric where one adds the vertical and horizontal distances. These gave the same topology, but the first has at most 3 equidistant points whereas the other has 4.

My question is whether this quantity has been studied and if it has a name. Also, can two metrics have an arbitrarily large difference in this number even though they induce the same topology? Can one have a finite number where the other has an infinite?

I am asking purely out of curiosity, by the way.

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These types of problems are studied for subsets of the Euclidean plane, but not in topology, since as you say the question is not topologically invariant; rather they are studied in combinatorial geometry. One classic problem along these lines is the Hadwiger-Nelson problem. Another question along these lines is the Erdős distance problem.

The answer to your second question is trivially yes. Consider the metric space $M_1$ on, say, countably many points where all distinct points are at distance $1$ from each other and the metric space $M_2$ on countably many points where all distinct points are at different distances from each other in, say, the interval $[0.5, 1]$. Both induce the discrete topology.

(These types of problems are also unlikely to be studied by anyone studying metric spaces, since it is only very rarely that anyone cares about metric spaces up to isometry. Most people only care about metric spaces up to homeomorphism, a few up to uniform isomorphism or quasi-isometry...)

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