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.