# Largest number of equidistant points

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.

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.