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I have spelled out the proof at paracompact Hausdorff spaces equivalently admit subordinate partitions of unity.
This uses Urysohn’s lemma and the skrinking lemma, whose proofs are not yet spelled out on the $n$Lab.
Okay, proofs of
are now all filled in, so now the proof at paracompact Hausdorff spaces equivalently admit subordinate partitions of unity is complete. I have added that it needs excluded middle and the axiom of choice (for the shrinking lemma).
If you use the axiom of choice, then you don't have to separately mention excluded middle, which follows from choice. (Excluded middle is Kuratowski-finite choice.) Of course, if you only use (say) countable choice, then excluded middle is independent of that.
I have tweaked the statement paracompact Hausdorff spaces equivalently admit subordinate partitions of unity so that it now reads: (Assuming the axiom of choice) Let $X$ be $T_1$. Then $X$ is paracompact and Hausdorff iff every open cover admits a subordinate partition of unity. I also tweaked the “if” part in the proof so as to derive Hausdorffness.
It’s just a slight strengthening of the statement, but a useful one. :-)
Thanks. I had been thinking I should do this, but didn’t get around to. Thanks very much.
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