Metrics with locally finite collections of balls
by
Gary Gruenhage
Auburn University
Coauthors: Zoltan Balogh (Miami University)

It is an old result of J. Nagata that every metrizable space admits a metric such that, for every \epsilon > 0, the collection of all \epsilon-balls is closure-preserving. More recently, Nagata asked if every metrizable space admits a metric such that the collection of \epsilon-balls (for any fixed \epsilon > 0) is locally finite. E.g., any separable metrizable space admits such a metric. However, we show that the answer is negative by characterizing the spaces which admit such a metric as precisely those spaces which embed in \kappa^\omega ×[0, 1]^\omega for some cardinal \kappa.

Date received: February 9, 2000