A closed surface is a surface that is compact and without boundary. Examples are spaces like the sphere, the torus and the Klein bottle. Examples of non-closed surfaces are: an open disk, which is a sphere with a puncture; a cylinder, which is a sphere with two punctures; and the Möbius strip.
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... The classification of closed surfaces has been known since the 1860s, and today a number of proofs exist ... proof of the classification is (Seifert Threlfall 1934), which brings every triangulated surface to a standard form ... This was originally proven only for Riemann surfaces in the 1880s and 1900s by Felix Klein, Paul Koebe, and Henri Poincaré ...