A sphere with a single cross-cap. This term is more appropriate in purely topological applications than the more common
term real projective plane, which implies
the presence of an affine structure (Francis and Weeks 1999). The double cross surface
is the Klein bottle and the triple cross surface
is called Dyck's surface (Francis and Collins 1993,
Francis and Weeks 1999).
Francis, G. and Collins, B. "On Knot-Spanning Surfaces: An Illustrated Essay on Topological Art." Ch. 11 in The
Visual Mind: Art and Mathematics (Ed. M. Emmer). Cambridge, MA: MIT
Press, 1993.Francis, G. K. and Weeks, J. R. "Conway's
ZIP Proof." Amer. Math. Monthly106, 393-399, 1999.