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Can we assign in a continuous manner to each point of the sphere $S^2$ a two point subset of S^2? I think this would contradict in some way "The Poincare theorem" Thanks

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What topology are you placing on the set of two-point subsets? –  Chris Eagle Oct 30 '12 at 19:21
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Not that it matters, since any constant function will be continuous anyway. –  Chris Eagle Oct 30 '12 at 19:22
    
So you want a map $S^2 \rightarrow S^2 \times S^2$? How about $f(x)=(x,0)$. –  JSchlather Oct 30 '12 at 19:24
    
You’re thinking, perhaps, of the canonical map from $S^2$ to ${\mathbb{P}}^2$? It is indeed continuous. –  Lubin Oct 30 '12 at 19:24
    
Jacob $0$ is not in $S^2$. –  Pedro Perez Oct 30 '12 at 19:49

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