All closed surfaces, despite their seemingly diverse forms, are topologically equivalent to spheres with some number of handles or cross-caps. The traditional proof follows Seifert and Threlfall (1980), but Conway's so-called "zero-irrelevancy" ("ZIP") provides a more streamlined approach (Francis and Weeks 1999).
Classification Theorem of Surfaces
See also
Cross-Cap, HandleExplore 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.Referenced on Wolfram|Alpha
Classification Theorem of SurfacesCite this as:
Weisstein, Eric W. "Classification Theorem of Surfaces." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ClassificationTheoremofSurfaces.html