In mathematics, the study of the properties of a geometric object that remains unchanged by deformations such as bending, stretching, or squeezing but not breaking. A sphere is topologically equivalent to a cube because, without breaking them, each can be deformed into the other as if they were made of modeling clay. A sphere is not equivalent to a doughnut, because the former would have to be broken to put a hole in it. Topological concepts and methods underlie much of modern mathematics, and the topological approach has clarified very basic structural concepts in many of its branches. See also algebraic topology.

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