Results which allow one to classify completely a collection of objects are among the most important and aesthetically pleasing in mathematics. The fact that they are also rather rare adds even more to their appeal. As specific examples, we mention the classification of finitely generated abelian groups up to isomorphism in terms of their rank and torsion coefficients; that of quadratic forms in terms of the rank and signature of a form; and that of regular solids up to similarity by the number of edges of each face and the number of faces meeting at each vertex. It should be clear that we have no hope of classifying topological spaces up to homeomorphism, or even up to homotopy equivalence. We can, however, give a complete classification of closed surfaces.
KeywordsEuler Characteristic Closed Surface Klein Bottle Simple Closed Curve Boundary Circle
Unable to display preview. Download preview PDF.