The completions of metric ANR's and homotopy dense subsets

摘要

In this paper, considering the problem when the completion of metric ANR X is an ANR and X is homotopy dense in the completion, we give some Sufficient conditions. It is also shown that each uniform ANR is homotopy dense in any Metric space containing X 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. Furthermore, introducing the notions of densely (local) Hyper-connectedness and uniformly (local) hyper-connectedness, we characterize of AR's (ANR's) and uniform AR's (uniform ANR's), respectively.