$f$ is a mapping: $(\mathbb{R}^2\setminus{0}) \rightarrow \mathbb{R}P^1$, which assigns to a point of the plane the line joining that point to the origin. The imbedding of $(\mathbb{R}^2\setminus{0})$ in $\mathbb{R}^2$ imbeds the graph of $f$ in the product $M= \mathbb{R}^2 \times \mathbb{R}P^1$. The closure of the graph in $M$ is called $S$. Prove that $S$ is diffeomorphic to a Möbius band.
Prove a surface to be diffeomorphic to a Möbius band.
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general-topology
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0It might help to think of M as a solid torus. – 2012-10-21
1 Answers
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You can take the open annulus $1
It's a good thing the overall dimension of $R^2$ X $RP^1$ is three, since I recall one cannot embed a strip into the plane. :-)
Not really sure this is rigorous, but it seems intuitive to me.
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0@azimut -- Any idea about a rigorous answer? I just dashed this off (many moons ago) and didn't think again about it. – 2013-05-06