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how do I show that there is homeomorphism $f:H^2\longrightarrow S^2$, where $H^2$ is the closed upper hemisphere with antipodal equator points identified and $S^2$ is the sphere with antipodal points identified?

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Any constant map is continuous. What kind of conditions do you want on this map? –  Qiaochu Yuan Feb 24 '12 at 18:48
    
I edited :) now I think is correct. –  Jr. Feb 24 '12 at 18:58
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There is one natural bijection between those spaces. It is enough to show the continuity of that. –  savick01 Feb 24 '12 at 19:03

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Well, note that a quotient space made from a compact space is compact. Consequently $H^2$ is compact. It is also Hausdorff. $S^2$ is Hausdorff, too. Any continuous bijection from a compact and Hausdorff space into a Hausdorff space is a homeomorphism. So it is enough to show that the natural bijection $f:H^2 \to S^2$ is continuous.

$f$ is the composition of the natural injection of a hemisphere (with quotient equator) into a sphere (with quotient equator) and the quotient map (from a sphere with quotient equator into a quotient sphere), so $f$ is continuous.

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what do you mean by "Sphere with quotient equador"? –  Jr. Feb 24 '12 at 19:24
    
@Jr.: sphere with antipodal equator points identified. –  savick01 Feb 24 '12 at 19:25
    
@savicko1: the quotient projection you mention send the whole equador to which point in the quotient sphere? –  Jr. Feb 24 '12 at 19:39
    
@Jr.: The second quotient is identifying all the non-equator antipodal points (equator points are already identified - I mean the antipodal ones). So on the equator with antipodal points identified the projection is just identity. Oh, I can see that my quotient projection is called quotient map in English. –  savick01 Feb 24 '12 at 19:44
    
@Jr.: Do you understand it now? Or should I elaborate on some point? I wanted to make it short and simple, but I might have gone too far. I use the fact that the topology you get when identifying some points in one go is the same as the topology you get when identifying them in a few steps (quotient of a quotient). It is an immediate proof from the definition of the quotient topology. –  savick01 Feb 24 '12 at 23:21

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