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From this paper enter link description here

$G$ denotes a discrete group. A $G$-CW-complex is a CW-complex upon which $G$ acts by permuting the cells. A $G$-CW-complex is said to be proper if each cell stabiliser is finite, which is equivalent to the requirement that $G$ should act properly discontinuously.

I wonder why proper $G$-CW complex implies $G$ should act properly discontinuously?

  • Every compact subset is contained in a finite union of cells. – Moishe Kohan Mar 11 '17 at 21:40
  • http://www.mathreference.com/at-cw,compact.html – Moishe Kohan Mar 11 '17 at 21:48
  • @MoisheCohen Thank you, but I need to show every point has an open neighbourhood $U$ such that $U\cap g(U)$ is nonempty for only finitely many $g\in G$. And a CW-complex is not necessarily locally compact, I don't know how to apply your words. –  Mar 11 '17 at 22:53
  • the two definitions of proper discontinuity are equivalent regardless of local compactness. Take a look at Bourbaki's General Topology. – Moishe Kohan Mar 11 '17 at 22:56
  • @MoisheCohen Thank you, I am not sure about your definition of "properly discontinuous", can you state it? –  Mar 11 '17 at 22:59
  • The natural map $ (G,X)\to (X,X)$ is proper. I highly recommend reading Bourbaki's or Koszul's book on transformation groups – Moishe Kohan Mar 11 '17 at 23:03
  • @MoisheCohen here you can see they are not generally equivalent http://math.stackexchange.com/questions/1082834/properly-discontinuous-action-equivalent-definitions –  Mar 11 '17 at 23:05
  • Oh, I see: You are using wrong definition of proper discontinuity (I did not notice that you have one open set, not two). I suggest, you avoid this definition as it is inadequate. The right definition in terms of neighborhoods is that given any two points $x, y\in X$ there are neighborhoods $U_x, U_y$ of these points such that ${g\in G: gU_x\cap U_y\ne \emptyset}$ is finite. This is the stronger definition and is equivalent to the one given in Bourbaki. – Moishe Kohan Mar 13 '17 at 15:58
  • @MoisheCohen Thank you very much! Actually, I just use the definition from Hatcher's book. And there are many Bourbakis, which one are you referring? –  Mar 13 '17 at 23:28
  • N. Bourbaki, "Elements of Mathematics. General Topology'', Part III. – Moishe Kohan Mar 13 '17 at 23:58
  • @MoisheCohen Thank you! –  Mar 14 '17 at 00:06
  • @Katherine: Have you figured out a proof? I also only see a way that Moishe Cohen's idea works if we additionally assume that $X$ is locally compact. – Timm von Puttkamer Dec 14 '17 at 08:48

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