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The completions of metric ANR’s and homotopy dense subsets. (English) Zbl 0974.57013

The author first gives some sufficient conditions when the completion \({\gamma}X\) of a metric ANR space \(X\) which is homotopy dense in \({\gamma}X\) is an ANR. He also shows that each uniform ANR is homotopy dense in any metric space in which it is contained isometrically as a dense subset and that a metric space \(X\) is a uniform ANR if and only if the metric completion of \(X\) is a uniform ANR with \(X\) a homotopy dense subset. Finally, introducing the notions of densely (local) hyper-connectedness and uniformly (local) hyper-connectedness, he characterizes ANR’s and uniform ANR’s respectively.

MSC:

57N20 Topology of infinite-dimensional manifolds
58D05 Groups of diffeomorphisms and homeomorphisms as manifolds
46E15 Banach spaces of continuous, differentiable or analytic functions
55M15 Absolute neighborhood retracts
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