All it is, is a compactification of the plane, not using the one-point "point at infinity" but by gluing in a 'slope' circle at infinity.
Heartbroken at the sheer awful complexity of Internet descriptions of the cross-cap. It's nearly the simplest possible manifold!
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i.e., two parallel lines intersect at the 'point at infinity' labeled by their slope. That's it. That's the crosscap.
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Wikipedia and Wolfram articles are TERRIBLE trainwrecks of wooliness.
End of conversation
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. Banned in Sweden. SubGenius, Zhuangist, white-hat troll. Defrocked mathematician. Brain problems.