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Reconstruction of manifolds and subsets of normed spaces from subgroups of their homeomorphism groups. (English) Zbl 1114.57023

This monograph considers the following situation. One is given a class of topological spaces and for each \(X\) in the class one is given a subgroup \( G(X)\) of the group of homeomorphisms of \(X.\) Being given two spaces in the class and an isomorphism of \(G(X)\) and \(G(X^{1})\) one wants a homeomorphism from \(X\) to \( X^{1}\) inducing the isomorphism. For example one could consider open subsets of a Banach space and homeomorphisms with \(f\) and \(f^{-1}\) both uniformly continuous and want the homeomorphisms from \(X\) to \(X^{1}\;\)and their inverses to be uniformly continuous.

MSC:

57N20 Topology of infinite-dimensional manifolds
46B99 Normed linear spaces and Banach spaces; Banach lattices
58B99 Infinite-dimensional manifolds
54E40 Special maps on metric spaces
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