## Journal of the Mathematical Society of Japan

### The completions of metric ANR's and homotopy dense subsets

Katsuro SAKAI

#### 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.

#### Article information

Source
J. Math. Soc. Japan, Volume 52, Number 4 (2000), 835-846.

Dates
First available in Project Euclid: 10 June 2008

https://projecteuclid.org/euclid.jmsj/1213107112

Digital Object Identifier
doi:10.2969/jmsj/05240835

Mathematical Reviews number (MathSciNet)
MR1774631

Zentralblatt MATH identifier
0974.57013

#### Citation

SAKAI, Katsuro. The completions of metric ANR's and homotopy dense subsets. J. Math. Soc. Japan 52 (2000), no. 4, 835--846. doi:10.2969/jmsj/05240835. https://projecteuclid.org/euclid.jmsj/1213107112