Approximation by zero-free continuous maps

TL;DR

This paper demonstrates that zero-free continuous maps on subsets of manifolds can be uniformly approximated by zero-free maps on the entire set, extending the approximation to the fine topology.

Contribution

It introduces a method to approximate zero-free maps on subsets of manifolds by zero-free maps on the whole set, enhancing understanding of map approximation in topology.

Findings

Zero-free maps can be approximated in the fine topology.

Approximation extends to uniform topology.

Method applies to subsets of n-dimensional manifolds.

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.