A note on ANRʼs

It is shown that if for a complete metric space ( X , d ) there is a constant ε > 0 such that the intersection ⋂ j = 1 n B d ( x j , r j ) of open balls is nonempty for every finite system x 1 , … , x n ∈ X of centers and a corresponding system of radii r 1 , … , r n > 0 such that d ( x j , x k ) ⩽ ε 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 = ∞ , the space X is an AR. A certain criterion for an incomplete metric space to be an A(N)R is presented.