We study the product X of discrete topological spaces  and give a necessary and sufficient condition for X to be discrete. Also, as an example of the product of discrete topological spaces we treat the set of configurations  of a cellular automaton  for the ring   of rational integers, the set S of states, the neighborhood N of 0 and the local function f, and show that the product space  is a compact metric space.

cellular automaton, configuration set, discrete topology, product of spaces.