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Klein bottle SAM.

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Klein bottle MAT.

kleinsche Flasche

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For example, the Klein bottle has no "boundary", where the surface stops abruptly, and it is non-orientable, as reflected in the one-sidedness of the immersion. en.wikipedia.org

This is an immersion of the Klein bottle in three dimensions. en.wikipedia.org

Gluing the circles together will produce a new, closed manifold without boundary, the Klein bottle. en.wikipedia.org

He makes the same restriction on geometrically atoroidal manifolds (which he calls topologically atoroidal) and in addition requires them to avoid incompressible boundary-parallel embedded Klein bottles. en.wikipedia.org

The Klein bottle, proper, does not self-intersect. en.wikipedia.org

Note that this is an abstract gluing in the sense that trying to realize this in three dimensions results in a self-intersecting Klein bottle. en.wikipedia.org

Gluing two projective planes creates the Klein bottle. en.wikipedia.org

This immersion is useful for visualizing many properties of the Klein bottle. en.wikipedia.org

Other examples of closed manifolds are the torus and the Klein bottle. en.wikipedia.org

Note that in three-dimensional space, a Klein bottle's surface must pass through itself. en.wikipedia.org

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