A note on ANR's

It is shown that if for a complete metric space $(X,d)$ there is a constant $ϵ>0$ such that the intersection ${\bigcap }_{j=1}^n{B}_d(x_j,r_j)$ of open balls is nonempty for every finite system $x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n\in X$ of centers and a corresponding system of radii $r_1,\mathrm{.}\mathrm{.}\mathrm{.},r_n>0$ 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 $ϵ=\mathrm{\infty }$, the space $X$ is an AR. A certain criterion for an incomplete metric space to be an A(N)R is presented.