h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

An amusing example of metrics is given by ultrametric distances, i.e. metrics d on a space X such that the triangle inequality is replaced by the stronger requirement d(x,y)<=max(d(x,z) ; d(y,z)) for all x,y,z. In such a space, it is a classical and amusing example to show that if you take an open ball of radius 1, say B, then any point b inside B is such that B is the ball of center b and radius 1 - in other words, any point in B is a center.

Perhaps the most classical example is in the following context. Take Q=set of rational numbers, p=prime number. For a nonzero rational number r, you can write r as product of primes with integral exponents, possibly negative; let N(r) be the exponent of p appearing in r. Now if a,b are two rational numbers, let d(a,b)=0 if a=b and d(a,b)=exp(-N(a-b)) otherwise. If my formula is correct, this should be a distance, called the "p-adic distance", which is ultrametric.

Another kinda classical example arises in the context of filtered vector spaces, much beyond the scope of this posting.