Suppose M is a countable metric space that contains at least two elements. Prove

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Let x_0 be an element of M. We know that the function f:M-->R defined as f(x)=d(x,x_0), where d is the metric of M, is a continuous function. Since M is countable the image of f is countable. Since a countable subset of R can't be an interval, as all intervals are uncountable, we infer that the image of f, say S, is disconnected subset of R. Since f was continuous, the inverse image of S under f is also disconnected. Since f^{-1}(S)=M, we get that M is disconnected.

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