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How would I construct the map? Once constucted, would I be right in saying that there is no Diffeomorphism to map back? As in $\mathbb{RP}^2$ a closed curve would have to have either $2$ points that represent the same $f(x)$ or one point that has $2$ values for $f(x)$? (It's not bijective)

Any advice is appreciated.

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    Easy: just map every point of $\Bbb{S}^2$ to the same point of $\Bbb{RP}^2$.2012-12-12
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    You are correct that the two are not diffeomorphic - one is orientable, and the other is not. It *may* be possible to find a smooth bijection $\mathbb{S}^2\to\mathbb{RP}^2$ though, it just won't have a smooth inverse. I'm not sure if such a thing exists though.2012-12-12
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    @MattPressland: A continuous bijection from a compact space to a Hausdorff space is a homeomorphism.2012-12-12
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    @ChrisEagle Ah, indeed. Very nice!2012-12-12
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    You can construct a map $f\colon S^2 \rightarrow RP^2$ easily. Just choose any point $p$ of $RP^2$ and define $f(x) = p$ for every $x \in S^2$. It is not only continuous, but also smooth. If you mean a bijective smooth map, there is no such thing because they are not homeomorphic.2012-12-12

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