Every infinite set admits a metric with no isolated points.

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# No point left behind

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Every infinite set admits a metric with no isolated points.

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

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Another solution: any infinite set is in bijection with . Give the discrete metric and the usual metric. Then the supremum metric on has no isolated point, and one may transport this to .

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