The completions of metric ANR's and homotopy dense subsets (Q1592428)

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.