Remove duplicate link to related statements

Stéphane Desarzens

]]>Fix typo

Stéphane Desarzens

]]>Ironically, I added the relevant references to the nLab article numerable open cover and forgot about them.

]]>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.

]]>I guess there’s this: https://mathoverflow.net/a/360635/, though that points back to the nLab.

]]>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.

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.

]]>Thanks. I had been thinking I should do this, but didn’t get around to. Thanks very much.

]]>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. :-)

]]>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.

]]>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).

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.

]]>