The completions of metric ANR's and homotopy dense subsets

Abstract

In this paper, considering the problem when the completion of a 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.