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, Handle## Explore 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 Surfaces## Cite this as:

Weisstein, Eric W. "Classification Theorem of Surfaces." From *MathWorld*--A Wolfram Web Resource.
https://mathworld.wolfram.com/ClassificationTheoremofSurfaces.html