## 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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