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Keywords:
hyperspace; Hausdorff metric and uniformity; metric manifold
hyperspace; Hausdorff metric and uniformity; metric manifold
Summary:
The local coincidence of the Hausdorff topology and the uniform convergence topology on the hyperspace consisting of closed graphs of multivalued (or continuous) functions is related to the existence of continuous functions which fail to be uniformly continuous. The problem of the local coincidence of these topologies on ${}C(X,Y)$ is investigated for some classes of spaces: topological groups, zero-dimensional spaces, metric manifolds.
References:
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[2] Deimling K.: Nonlinear Functional Analysis. Springer-Verlag Berlin (1985). MR 0787404 | Zbl 0559.47040
[3] Guillemin V., Pollack A.: Differential Topology. Prentice-Hall Inc. Englewood Cliffs NJ (1974). MR 0348781 | Zbl 0361.57001
[4] Isbell J.R.: Uniform Spaces. Mathematical Surveys nr 12 AMS Providence, Rhode Island (1964). MR 0170323 | Zbl 0124.15601
[5] Naimpally S.: Graph topology for function spaces. Trans. Amer. Math. Soc. 123 (1966), 267-272. MR 0192466 | Zbl 0151.29703
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