Cross-cap
In mathematics, a cross-cap is a two-dimensional surface in 3-space that is one-sided and the continuous image of a Möbius strip that intersects itself in an interval. In the domain, the inverse image of this interval is a longer interval that the mapping into 3-space "folds in half". At the point where the longer interval is folded in half in the image, the nearby configuration is that of the Whitney umbrella.
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The interval of self intersection precludes the cross-cap from being homeomorphic to the Möbius strip, but there are only two points in the image (the endpoints of the interval of self-intersection) where the image cannot be that of an immersion. The bounding edge of a cross-cap is a simple closed loop. Like certain versions of the Möbius strip, it may take the form of a symmetrical circle.
A cross-cap that has been closed up by gluing a disc to its boundary is a model of the real projective plane P2 (again with an interval of self-intersection, and two points where this model is not an immersion of P2).
Two cross-caps glued together at their boundaries form a model of the Klein bottle, this time with two intervals of self-intersection and four points where this model is not an immersion.
An important theorem of topology, the classification theorem for surfaces, states that each two-dimensional compact manifold without boundary is homeomorphic to a sphere with some number (possibly 0) of "handles" and 0, 1, or 2 cross-caps.