A note on ANR's

TL;DR

The paper establishes conditions under which a complete metric space is an ANR or AR based on intersection properties of open balls with bounded or unbounded centers, and provides a criterion for incomplete spaces.

Contribution

It introduces new intersection-based criteria for identifying ANRs and ARs in metric spaces, including incomplete spaces.

Findings

Complete metric spaces with bounded intersection properties are ANRs.

Unbounded intersection conditions imply the space is an AR.

A criterion for incomplete spaces to be ANRs or ARs is provided.

Abstract

It is shown that if for a complete metric space $(X,d)$ there is a constant $\epsilon > 0$ such that the intersection $\bigcap_{j=1}^n B_d(x_j,r_j)$ of open balls is nonempty for every finite system $x_1,...,x_n \in X$ of centers and a corresponding system of radii $r_1,...,r_n > 0$ such that $d(x_j,x_k) \leqsl \epsilon$ and $d(x_j,x_k) < r_j + r_k$ ($j,k = 1,...,n$), then $X$ is an ANR; and if in the above one may put $\epsilon = \infty$, the space $X$ is an AR. A certain criterion for an incomplete metric space to be an A(N)R is presented.