The compact countable metric spaces are topologically classified simply
by the classical Mazurkiewicz-Sierpi\'{n}ski theorem. Our concern is
non-compact case. After viewing the scattered countable metric spaces
of length 2 and the locally compact countable metric spaces, we shall
prove Theorem 2, the main theorem of the present paper. Theorem 2
presents a topological classification of a class of scattered countable
metric spaces which is far from the class of locally compact countable
metric spaces.