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My question is about the following non-referenced claim on the page: https://en.wikipedia.org/wiki/Partition_of_unity

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Is it really true? Where can I find a precise reference of this property?

André Porto
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    The 'claim' is not true, just very badly written. I think it's trying to define something, but its so awfully put together that I can't make heads or tails of what the definition is meant to be. My advice: disregard wikipedia and pick up a good topology text book. – Tyrone Aug 03 '20 at 15:32
  • Do you have a sugestion for topology textbook on partitions of unity? I already searched at Engelking with no success. – André Porto Aug 03 '20 at 20:28
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    Engelking's book is very good. Obviously the key word is paracompactness. Other books I like are Nagata's Modern General Topology, James's General Topology and Homotopy Theory and Mukres's Topology. Also you will find some useful facts in the appendix of Dold's book Lectures on Algebraic Topology, and in tom Dieck's Algebraic Topology. For information about the smooth picture you can do no better than Hirsch's Differential Topology. – Tyrone Aug 03 '20 at 20:42
  • By the way, here is constructed a locally compact Hausdorff space which is not paracompact. – Tyrone Aug 03 '20 at 20:43
  • I really apreciated the references. I understand that there are locally compact $T_2$ that are not paracompact, even if we require it to be normal. However, even though that, at least for $T_2$ spaces, being paracompact is equivalent to admiting a subordinated partition of unity for any open cover, the property stated is weaker than being subordinated, since it allows the index set of the functions to be different from the index set of the open cover. Maybe one of the key questions is whether the property stated implies the existence of locally finite open refinements for any open cover of X. – André Porto Aug 03 '20 at 21:06
  • The assumptions are sufficient to guarantee that the associated cover is numerable: any open cover of a space $X$ which is refined by the cozero sets of some partition of unity is precisely subordinated by a locally finite partition of unity. See Engelking pg 301 onwards. Note his terminology (and mine) is not standard. Details are also found in Dold's text (pg. 352 onwards), James's book and tom Dieck's book (Le. 13.1.2 onwards) – Tyrone Aug 03 '20 at 21:23
  • Uh, nice one! Now I got it. The fact you said is precisely Lemma 5.1.8. It really had to do with the terminology. I was thinking that Engelking's "subordinated" required the index sets to be equal. Now I have to get in contact with wikipedia to avoid further messes around this topic. I appreciate the patience. – André Porto Aug 03 '20 at 22:16

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