Can you create a topological metric space from three points (the set X) in the plane? Obviously, d is defined.

A ball at each point is the point itself (satisfies def of neighborhood of a point).

Union of all the balls is X.

Intersection of any two balls is empty (contains no member of X) so there can't be an x in a ball which is a subset of the intersection.

Is it valid to conclude you can't create a topolgical metric space from a finite set consisting of three or more members? Or does a topological metric space have to have an infinite number of members?

EDIT Origin of the question: A metric is defined on a non-empty set, which could have a finite number of members. In a std text developments go on from there without ever considering the possibility that the set is finite (three or more members to satisfy d), leaving me scratching my head.