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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 shrinking 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.
Why exactly do we need T1 and Hausdorffness properties here?
Is it not true that paracompact topological spaces are precisely those spaces that admit a subordinate locally finite partition of unity for every open cover?
The proofs and their dependencies appear to make no use of T1 or Hausdorffness properties.
Can you get away from using Urysohn’s lemma? That requires the space to be normal, but maybe you mean the version without $T_1$?
It’s a pity Bourbaki in General Topology defines “paracompact” to include the Hausdorff condition, otherwise that would be a good place to check.
I guess there’s this: https://mathoverflow.net/a/360635/, though that points back to the nLab.
Re #8: Curiously, it points back to the nLab article for which I wrote the cited section.
So it appears that the correct condition is that the space is a fully normal topological space.
Ironically, I added the relevant references to the nLab article numerable open cover and forgot about them.
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