# Real projective plane and Möbius strip

Let $\mathbb{S}^2$ be the unit sphere in $\mathbb{R}^3$. Define the quotient space $$\mathbb{R}P^2:=\mathbb{S}^2/(x\sim -x).$$

How can we show that $\mathbb{R}P^2$ contains an embedding of the Möbius strip

$$M:=([0,1]\times (-1,1))/((0,y)\sim (1,1-y))$$

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For example, the Möbius strip can map to a thin belt around the equator of the sphere. Then the relation between this belt (in spherical coordinates!) and your defining parameterization of the Möbius strip is a simple affine transformation of each coordinate separately. Remember that $[0,1]$ should map to only one hemisphere of longitude due to the identification of antipodes.