TITLE: Is $\mathbb{R} \backslash \mathbb{Q}$ locally compact?
QUESTION [0 upvotes]: So I believe the answer is no, because the answer is no for rationals. I feel that every word in the proof of rationals not being locally compact can be changed to irrationals. But I wasn't totally confident since I am a beginner student of topology.

REPLY [1 votes]: You're right, $\Bbb P$ (the irrationals) are a non-locally compact space, and the proof can be similar to a proof as given for $\Bbb Q$ (but for the latter we can also say it's not Baire so cannot be locally compact, but $\Bbb P$ is actually Baire so that particular argument does not translate).
You can use that irrational sequences can converge to rational points, e.g.
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