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A topological space is totally disconnected if the connected components are all singletons. Prove that any...

A topological space is totally disconnected if the connected components are all singletons. Prove that any countable metric space is totally disconnected.

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Answer #1

Every countable metric space X is totally disconnected

proof:- proof is simple

Assume that X is countable

let xX , then the set

D={d(x,y) : yX } is countable

thus there exists rn →0 with rnD

Then B(x,rn) is both open and closed . Since the sphere of radius rn about x is empty . Thus the largest connected set containing x is x itself so by definition X is totally disconnected

​​​ The converse however is not true

A counter example is Q

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