For a finite cell decomposition of the sphere into polygonal faces, the numbers of vertices, edges and faces satisfy the displayed formula. Triangulating faces preserves because each added diagonal adds one edge and one face. This is the spherical form of Euler formula for a connected planar graph, where the plane's exterior face is included.
For finite CW complexes, the product cell structure has one -cell for every pair consisting of a -cell and a -cell. Taking the alternating cell count gives
If is a finite covering of degree between spaces having finite CW type, thenA lifted CW structure has exactly cells above each cell of the base.
Every closed odd-dimensional manifold has Euler characteristic zero. For an orientable manifold this follows by pairing complementary Betti numbers using Poincare duality; the general case follows from the orientable double cover and Euler characteristic under a finite covering.
If a finite simplicial complex has simplices of dimension , thenWriting a lift of and makes the boundary dimensions cancel in the alternating sum.
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The Euler characteristic is a topological invariant that describes a fundamental property of a topological space. It is often denoted by the Greek letter \( \chi \) (chi) and is defined for various types of spaces, including polyhedra, surfaces, and more generally, topological spaces.