Approximation by zero-free continuous maps

Abstract

We prove that if E a subset of an n-dimensional manifold, then every continuous R^n-valued map on E that is zero-free on the interior of E can be approximated in the fine topology, and hence, in particular, in the uniform topology, by a continuous R^n-valued map that is zero-free on all of E.

The main theorem has been strengthened in two ways: (i) The theorem now applies to subsets of an n-manifold rather than just subsets of Euclidean n-space. (ii) The conclusion has been strengthened from approximation in the uniform topology to approximation in the fine topology