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In the Engelkings book "General Topology" in the chapter of paracompactness there is a lemma related to partitions of unity and locally finite covers.

I am trying to understand the proof of the lemma, but I am blocked at the part when $V_s$ sets are defined. I can't get why $V_s$ are open, why $\mathcal{V}$ is a refinement of $\mathcal{U}$ and why $\mathcal{V}$ is locally finite.

The lemma together with the proof is given below:enter image description here.

Emo
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  • What is Engelking's notion of "a partition of unity subordinated to an open cover"? If it is the same as I would call thus, the proof is needlessly complicated, $W_s = {x \in X : f_s(x) > 0}$ would give a locally finite refinement without much ado. – Daniel Fischer Nov 08 '20 at 21:40
  • A partition of unity ${ f_s(x) }{s\in S}$ is subordinated to an open cover $\mathcal{A}$ if the cover ${ f^{-1}_s((0,1]) }{s\in S}$ is a refinement of $\mathcal{A}.$ – Emo Nov 08 '20 at 21:47
  • And does "partition of unity" imply that the supports are a locally finite family for Engelking? – Daniel Fischer Nov 08 '20 at 21:49
  • In this literature the notion of support is not used! – Emo Nov 08 '20 at 21:51
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    I see. That's why the simpler construction above doesn't work in Engelking's setting, he uses a wider definition of partition of unity. – Daniel Fischer Nov 08 '20 at 21:58
  • Okay, so if we write $V_s = {x \in X : f_s(x) - \frac{1}{2}f(x) > 0}$, can you see why that is an open set? – Daniel Fischer Nov 08 '20 at 22:03
  • I still didn't get it, until the above answer was posted :) – Emo Nov 08 '20 at 22:22

1 Answers1

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For $s\in S$ let $h_s(x)=f_s(x)-\frac12f(x)$; $h_s$ is continuous (because $f_s$ and $f$ are), so

$$V_s=\{x\in X:h_s(x)>0\}=h_s^{-1}\big[(0,\to)\big]$$

is open. For each $x\in V_s$ we have $\color{red}{f_s(x)>\frac12f(x)}>0$ by the definition of $f$, so $x\in f_s^{-1}\big[(0,1]\big]$. The partition of unity is subordinated to $\mathscr{U}$, so by definition there is a $U\in\mathscr{U}$ such that $V_s\subseteq f_s^{-1}\big[(0,1]\big]\subseteq U$, and $\mathscr{V}$ is a refinement of $\mathscr{U}$.

Finally, let $g=\frac12f$, and let $x\in X$; $f(x)>0$, so $g(x)>0$, and therefore there are a nbhd $U_0$ of $x$ and a finite $S_0\subseteq S$ such that $f_s(y)<g(y)=\frac12f(y)$ for all $y\in U_0$ and $s\in S\setminus S_0$. Let $s\in S\setminus S_0$ and $y\in U_0$; then $f_s(y)<\frac12f(y)$, so $y\notin V_s$ (by the red inequality above). Thus, $U_0\cap V_s=\varnothing$ for all $s\in S\setminus S_0$, so $\{s\in S:U_0\cap V_s\ne\varnothing\}\subseteq S_0$, and $U_0$ is therefore a nbhd of $x$ that meets only finitely many members of $\mathscr{V}$. Since $x\in X$ was arbitrary, this shows that $\mathscr{V}$ is locally finite.

Brian M. Scott
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  • This lemma is due to C. H. Dowker? Any references? – Mehmet Onat Apr 17 '23 at 13:59
  • @MehmetOnat: I don’t know about the lemma, but Engelking says that this proof of it is due to M.R. Mather in a 1964 preprint Paracompactness and partitions of unity. – Brian M. Scott Apr 17 '23 at 19:47
  • On page 213. https://www.google.com.tr/books/edition/Introduction_to_Topology/UQiZDwAAQBAJ?hl=tr&gbpv=1&dq=We+remark+that+the+converse+of+the+preceding+theorem+(due+to+C.+H.+Dowker&pg=PA213&printsec=frontcover. But no references – Mehmet Onat Apr 17 '23 at 19:52
  • Actually, I want to prove that if for an open cover $\mathcal{U}$ of a space $X$ there exists a partition of unity $\left{ f_{i}\right} _{i\in I}$ subordinated to it then $X$ is a paracompact space. – Mehmet Onat Apr 17 '23 at 19:56
  • Michael's article note on paracompact spaces, (Proposition 2) In short, he proves it. – Mehmet Onat Apr 17 '23 at 20:02
  • Is the book of M. R. Mather available online? – Mehmet Onat Apr 17 '23 at 20:04
  • @MehmetOnat: I don’t know whether the Mather paper was even published: Engelking mentions only a preprint of it. – Brian M. Scott Apr 17 '23 at 20:52
  • It is referred to as Mather, Paracompactness and partitions of unity, Mimeographed Notes 1964; PhD thesis,. Cambridge University, 1965 in literature. – Mehmet Onat Apr 17 '23 at 21:10
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    @MehmetOnat: So the mimeographed notes must be the preprint that Engelking mentions and must contain material that went into Mather’s PhD thesis. So far as I can discover, that was never published and does not appear anywhere online. – Brian M. Scott Apr 17 '23 at 21:48