A note on ANR's

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.