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Topologies induced by metrics with disconnected range

Published online by Cambridge University Press:  17 April 2009

Kevin Broughan
Kevin Broughan
Affiliation:
Department of Mathematics, University of Waikato, Private Bag, Hamilton, New Zealand.
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Abstract

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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.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 25 , Issue 1 , February 1982 , pp. 133 - 142
Copyright
Copyright © Australian Mathematical Society 1982

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