On isomorphisms with small bound

If X and Y are locally compact Hausdorff spaces, and if we denote by Co(X) and CO( Y) the Banach spaces of continuous, complexvalued functions vanishing at infinity on X and Y respectively, then, according to the Banach-Stone Theorem, the existence of an isometry between the function spaces Co(X) and Co(Y) implies that X and Y are homeomorphic. In [1 ] it was shown that if we assume, in addition, that X and Y satisfy the first axiom of countability, then this theorem can be generalized by considering isomorphisms with small norm instead of isometries. The object of this paper is to show that the conclusions of [1] remain valid without the assumption of first countability.