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Oct 26, 2009 at 18:06 comment added Thomas Sauvaget Yeah, I figured out after asking, so really the LS category measures something different, thanks!
Oct 26, 2009 at 17:53 comment added j.c. Note that the two open sets in Reid's answer may not be made contractible in general. Indeed, for orientable 2-manifolds, 3 contractible sets are required, except for S^2, where 2 are required.
Oct 26, 2009 at 16:54 history answered Reid Barton CC BY-SA 2.5