Cross-cap

In mathematics, a cross-cap is a two-dimensional surface that is topologically equivalent to a Möbius strip. The term 'cross-cap', however, often implies that the surface has been deformed so that its boundary is an ordinary circle. This cannot be done in three dimensions, without the surface intersecting itself.

A cross-cap that has been closed up by gluing a disc to its boundary is called a real projective plane. Two cross-caps glued together at their boundaries form a Klein bottle. An important theorem of topology, the classification theorem for surfaces, states that all two-dimensional nonorientable manifolds are spheres with some number of 'handles' and at most two cross-caps.