Locally injective maps in o-minimal structures without poles are surjective

Abstract

If f : R m → R m f:\mathbf R^m\to \mathbf R^m is continuous and locally injective, then f f is in fact surjective and a homeomorphism, provided f f is definable in an o-minimal expansion without poles of the ordered additive group of real numbers; ‘without poles’ means that every one-variable definable function is locally bounded. Some general properties of definable maps in o-minimal expansions of ordered abelian groups without poles are also established.