No point left behind

Every infinite set admits a metric with no isolated points.

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No point left behind

2 thoughts on “No point left behind

  1. Every infinite set is a disjoint union of countable sets. Each of these sets is in bijection with the rational numbers between 0 and 1. Glue these metrics together by declaring the distance between points in different sets to be 1.

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  2. Another solution: any infinite set X is in bijection with X\times\Bbb{Q}. Give X the discrete metric and \Bbb{Q} the usual metric. Then the supremum metric on X\times\Bbb{Q} has no isolated point, and one may transport this to X.

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