Consider the set of rational numbers, under the usual metric d(x,y)=|x-y|(adsbygoogle = window.adsbygoogle || []).push({});

I am pretty sure that this space is totally disconnected, but I can't convince myself that the set {x} U {y} is a disconnected set.

It seems obvious, but I can't find two non-empty disjoint open sets U,V such that U U V = {x} U {y}.

I am sure {x},{y} are not open sets, so I need something bigger.

Is the total disconnection only true if the the set Q is a relative topology of R?

Thanks

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# The topology of rational numbers: connected sets

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