Topologies induced by metrics with disconnected range

Abstract

In a metric space (X, d) a ball B(x, ε) is separated if d(B(x, ε), X\B(x, ε)] > 0. If the separated balls form a sub-base for the d-topology then Ind X = 0. The metric is gap-like at x if dx(X) is not dense in any neighbourhood of 0 in [0, ∞). The usual metric on the irrational numbers, P, is the uniform limit of compatible metrics (dn), each dn being gap-like on P. In a completely metrizable space X if each dense Gδ is an Fσ then Ind X = 0. © 1982, Australian Mathematical Society. All rights reserved.