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$f$ 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.

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    It might help to think of M as a solid torus.2012-10-21

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