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Suppose X is countable. We need to show that for any metric d on X the space (X,d) is totally disconnected.

It is true that any subset of a countable set is countable. so, divide the space until its components such a way it is the element we use for counting. thus, X is totally disconnected.

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Use 2 facts :

  1. Any connected metric space with at least 2 points is uncountable. This has been asked a few times here on MSE (see this, for instance)

  2. Any subset of a countable space is countable.

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