Journal of the Mathematical Society of Japan

The completions of metric ANR's and homotopy dense subsets

Katsuro SAKAI

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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

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

First available in Project Euclid: 10 June 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57N20: Topology of infinite-dimensional manifolds [See also 58Bxx] 58D05: Groups of diffeomorphisms and homeomorphisms as manifolds [See also 22E65, 57S05] 46E15: Banach spaces of continuous, differentiable or analytic functions

ANR homotopy dense the metric completion uniform ANR densely locally hyper-connected uniformly locally hyper-connected


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.

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