In geometry, topology concerns the global features of a shape, independent of the detail—a famous example being that a coffee mug and a torus are topologically equivalent because they can be smoothly transformed into each other without experiencing dramatic changes, e.g., opening holes, tearing, gluing. The Euler characteristic is introduced to describe such a global invariant, which is defined by the integral of the Gaussian curvature over a closed surface, , which is always an integer. Hence, it cannot vary continuously and is topologically stable. The surfaces of a sphere () and torus () are distinguished topologically by their Euler characteristics .
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