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in mathematics, a correspondence between two figures or surfaces or other geometrical objects, defined by a one-to-one mapping that is continuous in both directions. The vertical projection shown in the sets up such a one-to-one correspondence between the straight segment x and the curved interval y. If x and y are topologically equivalent, there is a function h:xy such that h is continuous, h is onto (each point of y corresponds to a point of x), h is one-to-one, and the inverse function, h1, is continuous. Thus h is called a homeomorphism.