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?
Real Projective Plane copies
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general-topology
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1Any constant map is continuous. What kind of conditions do you want on this map? – 2012-02-24
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0I edited :) now I think is correct. – 2012-02-24
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1There is one natural bijection between those spaces. It is enough to show the continuity of that. – 2012-02-24
1 Answers
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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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0what do you mean by "Sphere with quotient equador"? – 2012-02-24
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0@Jr.: sphere with antipodal equator points identified. – 2012-02-24
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0@savicko1: the quotient projection you mention send the whole equador to which point in the quotient sphere? – 2012-02-24
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0@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. – 2012-02-24
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0@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. – 2012-02-24
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0Sorry, I didn't understand the part: why the 2 functions you cited are continous? More generally,given a function $f:A \longrightarrow \frac{B}{\varstar}$, where $A$ is a space and $\frac{B}{\varstar}$ is a quotient space,how to describe the continuity of $f$ in terms of a simple function $g: A \longrightarrow B$ ? – 2012-02-26
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0The first is continuous because it is an injection. The second is continuous because it is a quotient map. More precisely: the quotient topology may be defined as the finest topology on the quotient space such that the quotient map is continuous (http://en.wikipedia.org/wiki/Quotient_space). General rule: if a function $A \to B/\simeq$ can be described as a composition $A \to B \to B/\simeq$ of a continuous function and the quotient map, then it is continuous. But it's not equivalence - there are continuous $f:A\to B/\simeq$ that cannot be composed in that way. – 2012-02-26
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0@savicko1 I got it now,thanks for the comments! – 2012-02-27