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It is a known fact that real numbers are locally compact and rationals are not with respect to the topology inherited from R.

What about irrationals?are they locally compact?

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Nope, the irrationals are not locally compact. What can you say about compact sets in the topology on the irrationals? What would happen if such a compact set contained an open neighborhood of a point? (Note that the irrationals are Hausdorff, so compact would imply closed).