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High Quality Content by WIKIPEDIA articles! The Vietoris Begle mapping theorem is a result in the mathematical field of algebraic topology. It is named for Leopold Vietoris and Edward G. Begle. The statement of the theorem, below, is as formulated by Stephen Smale. Let X and Y be compact metric spaces, and let f:Xto Y be surjective and continuous. Suppose that the fibers of f are acyclic, so that tilde H_r(f^{-1}(y)) = 0, for all 0leq rleq n-1 and all yin Y, with tilde H_r denoting the rth reduced homology group. Then, the induced homomorphism, f_ :tilde H_r(X)totilde H_r(Y), is an isomorphism for rleq n-1 and a surjection for r = n.