Approximation of maps into spheres by piecewise-regular maps of class C^k

TL;DR

This paper proves that continuous maps from compact subsets of real algebraic varieties into spheres can be approximated by piecewise-regular maps of any specified smoothness class C^k.

Contribution

It establishes a general approximation result for maps into spheres using piecewise-regular maps of arbitrary smoothness class C^k.

Findings

Any continuous map can be approximated by piecewise-regular C^k maps.

The approximation holds for maps from compact subsets of real algebraic varieties.

The result applies for any integer k, indicating arbitrary smoothness levels.

Abstract

The aim of this paper is to prove that every continuous map from a compact subset of a real algebraic variety into a sphere can be approximated by piecewise-regular maps of class C^k, where k is an arbitrary integer.