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The Klein bottle is a compact non-orientable surface (and hence, in particular, a connected two-dimensional manifold) defined in the following equivalent ways (up to homeomorphism)
- It is the connected sum of two copies of the real projective plane.
- It is obtained by taking a torus, removing one of the factor circles, and re-gluing this circle with the opposite orientation.
(More definitions, more precise definitions needed).
The Klein bottle is one of the compact non-orientable surfaces.
Further information: homology of Klein bottle
Further information: cohomology of Klein bottle
Further information: homotopy of Klein bottle