1. In this paper we shall s tudy A1NR's (absolute neighborhood retracts). The general problem will be as follows. Suppose we have proved that all ANR's have a certain property. Then we may ask, if this proper~ty is characteristic for ANR%, or in other words if it is true tha t a separable metric space having this property necessarily is an ANR. Thus we shall s tudy necessary conditions for a space to be an ANR, and we shall find tha t some of these conditions are also sufficient. Using KURATOWSKI'S modification ([7] p. 270) of BORSUK'S original definition ([1] p. 222), we mean by an ANR a separable metric space X such that, whenever X is imbedded as a closed subset of another separable metric space Z, it is a retract of some neighborhood in Z. First, we take up the study of local properties of ANR's. I t is known tha t an ANR is locally contractible (cf. [4] p. 273) and BORSUK proved tha t local contractability is sufficient for a finite dimensional compact space to be an ANR ([1] p. 240). In a recent paper, however, he has given an example of a locally contractible infinite dimensional space, which is not an ANR [3]. So the question then arises, if the property of a space to be an ANR is a local property. That the answer is affirmative is shown by theorem 3.3. In the case of a compact space this has already been proved by YAJIIMA [10]. Thereafter we prove some theorems on homotopy of mappings into an ANR. Briefly the result can be stated by saying that two mappings of the same space into an ANR which are "near" enough to each other, are homotopic, and tha t if the homotopy is already given on a closed subset and is "small" enough, then this homotopy is extendable. For a compact ANR we can give an exact meaning to the words near and small in terms of some metric. But the uniformity structure implied by a metric does not seem to be a suitable tool for handling non-compact ANR's. Instead of a metric we therefore use open coverings of the space. BORSUK proved [2] tha t any compact ANR X is dominated by a finite polyhedron P. This means tha t there exists two mappings q : X -+ P and ~v : P -+ X such tha t ~o ~0 : X -+ X is homotopic to the identity mapping i : X ~ X. We now prove tha t the polyhedron P and the mappings ~ and yJ can be chosen so tha t this homotopy between ~ ~ and i is arbitrarily small, and we show tha t in this way we get a sufficient condition. This result is generalized in a natural way to n?n-compact spaces by using infinite locally finite polyhedra. Since these polyhedra are ANR's (see corollary 3.5) we thus see tha t any ANR is dominated b y a locally compact A~R.