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

Lewenberg, Adam

doi:10.1090/S0002-9939-96-03352-7

Locally injective maps in o-minimal structures without poles are surjective
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by Adam H. Lewenberg

Proc. Amer. Math. Soc. 124 (1996), 2839-2844

DOI: https://doi.org/10.1090/S0002-9939-96-03352-7

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Abstract:

If $f:\mathbf R^m\to \mathbf R^m$ is continuous and locally injective, then $f$ is in fact surjective and a homeomorphism, provided $f$ 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.

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