Slightly more obscure (although in some ways, more important) examples include:
Discrete Metric Space
formed by taking any set and the distance function:
d(x,y) = 0 if x = y
d(x,y) = 1 if x != y
often used in pathological examples to prove various theorems and statements wrong.
Induced Metric
If A is a subset of X and (X,d) is a metric space, then we call (A,d) the induced metric on A. Whilst this seems obvious (if you do Math, anyway) it's very useful in proving theorems in other areas, such as Topology.