Invariant metrics in groups (solution of a problem of Banach)

Abstract

Introduction.If G is a semi-group and p a metric on G, p will be called left invariant if p(gx, gy) =p(x, y) whenever {g, x, y} CG, right invariant if always p(xg, yg) =p(x, y), and invariant if it is both right and left invariant.If T is a topological space and p a metric on T, we shall say that T admits p if the p-topology of T agrees with its original topology.G. Birkhoff [2]2 and Kakutani [5] proved that a Hausdorff group admits a left invariant metric if and only if it satisfies the first axiom of countability.§1 below contains some remarks on invariant metrics,3 including a slight sharpening of the theorem just mentioned.A topological space will be called topologically complete if it admits a metric under which it is complete.The principal result of this note (2.4) is that if G is a Hausdorff group which is abelian, metrizable, and topologically complete, then G admits an invariant metric under which it is complete.As applied to linear metric spaces, this answers affirmatively a question of Banach [l, p. 232].