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From the Wikipedia article on projective planes:

[...] consider the unit sphere centered at the origin in $\mathbb{R}^3$. Each of the $\mathbb{R}^3$ lines in this construction intersects the sphere at two antipodal points. Since the $\mathbb{R}^3$ line represents a point of $\mathbb{RP}^2$, we will obtain the same model of $\mathbb{RP}^2$ by identifying the antipodal points of the sphere. The lines of $\mathbb{RP}^2$ will be the great circles of the sphere after this identification of antipodal points.

My question is:

When the construction of the real projective plane is essentially identifying antipodal points of the sphere, what is its analogue when identifying antipodal points of the torus?

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    Note, however, that whatever choice you make you'll have to end u$p$ with either a Klei$n$ bottle or a torus -- you'll get a surface that has half the Euler characteristic of the torus, the torus has Euler characteristic zero, and the only other surface with Euler characteristic zero is the Klein bottle.2012-12-06

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It is not absolutely clear what antipodal points on a torus are. However, if you take a standard torus in $\mathbb{R}^3$, obtained by rotating a circle $(x-a)^2+z^2 = b^2$ with $a>b>0$ about the $z$-axis, and if you call $(-x,-y,-z)$ the antipodal point of $(x,y,z)$, I believe you get the Klein bottle.

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    @Aaron: Thanks - explicitly - for this hint.2012-12-06