A closed surface is a 2-dimensional compact manifold without boundary. Every connected closed surface is classified as either an orientable surface or a nonorientable surface and then by its genus: it is homeomorphic either to a sphere with some number of handles or to a sphere with some number of cross-caps. The sphere, torus, and real projective plane are standard examples.
Closed Surface
See also
Classification Theorem of Surfaces, Closed Manifold, Compact Surface, Nonorientable SurfaceExplore with Wolfram|Alpha
References
Francis, G. K. and Weeks, J. R. "Conway's ZIP Proof." Amer. Math. Monthly 106, 393-399, 1999.Seifert, H. and Threlfall, W. A Textbook of Topology. New York: Academic Press, 1980.Cite this as:
Weisstein, Eric W. "Closed Surface." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ClosedSurface.html