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Dec 23, 2010 at 3:05 comment added Dick Palais @Hugo Chapdelaine. It's in my Annals article, but that may be hard to get. It is also in an article by Abels, Manousssos, and Noskov called "Proper actions and proper invariant metrics" that is available as pdf online (see Proposition 2.3, page 4).
Dec 23, 2010 at 2:20 vote accept Hugo Chapdelaine
Dec 23, 2010 at 2:19 comment added Hugo Chapdelaine Thanks @Dick. So I agree that the equivalence between $X$ being a proper $G$-space iff the map $G\times X\rightarrow X\times X$ is proper not only answers Q1 but generalizes it. So where can I find a proof of this equivalence?
Dec 23, 2010 at 1:45 history answered Dick Palais CC BY-SA 2.5