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I was going to be lazy and follow Akhil Mathew’s proof here. But now I see that I don’t follow where he wrote (way back): “I claim that the $V_\beta$ are locally finite.”
In any case, that discussion should be at Michael’s theorem instead.
So I have added at second-countable regular spaces are paracompact the statement and the easy proof as a corollary of Michael’s theorem.
Re #1: this is in the proof of Lemma 3? Is there a more precise spot in that paragraph that could can point to where you’re stuck?
Sorry, yes. At the very end of that paragraph: The $G_\kappa$ are a closed cover. To draw the conclusion at the end of the paragraph, would we not need that their interiors still cover?
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