Every compact surface without boundary is homeomorphic to a sphere, a connected sum of tori, or a connected sum of projective planes, and is fully determined up to homeomorphism by its orientability and its Euler characteristic. A foundational result of low-dimensional topology, established over the late nineteenth and early twentieth centuries.
Facts
StatementThe classification theorem for closed surfaces states that every connected closed surface is homeomorphic to the sphere, to a connected sum of one or more tori, or to a connected sum of one or more real projective planes, so that Euler characteristic together with orientability completely determines a closed surface up to homeomorphism. 1 Classification
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Sources
1. Classification of surfaces (Wikipedia)
Wikimedia FoundationSurface (topology), Classification of closed surfaces section
The classification theorem of closed surfaces states that any connected closed surface is homeomorphic to some member of one of these three families:
In Group: Geometric and Differential Topology, Lead paragraph
In topology, a surface is a two-dimensional manifold.
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