consider the following group of isometries:

G = {T_{n}: n in Z} where T_{n}(x,y) = (x+n,y).

how close can (x+n,y) and (x,y) be?

(G can be thought of as modelling an "asymmetric (infinitely) repeating pattern", which is 1 unit long).

the "discreteness" of G essentially derives from the discreteness of Z in R (under the usual metric topology for the real line).