Cross-cap

In mathematics, a cross-cap is a two-dimensional surface that is a model of a Möbius strip with a single self intersection. This self intersection precludes the cross-cap from being topologically equivalent (i.e., homeomorphic) 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.

A cross-cap that has been closed up by gluing a disc to its boundary is an immersion of the 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 compact manifolds without boundary are homeomorphic to spheres with some number of ‘handles’ and at most two cross-caps.