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Topological definition

The polyhedral surfaces discussed above are, in modern language, two-dimensional finite CW-complexes. (When only triangular faces are used, they are two-dimensional finite simplicial complexes.) In general, for any finite CW-complex, the Euler characteristic can be defined as the alternating sum

where k denotes the number of cells of dimension n in the complex.

Similarly, for a simplicial complex, the Euler characteristic equals the alternating sum

where k denotes the number of n -simplexes in the complex.