American Journal of Mathematics

Abstract:

Let $Y\subset\Bbb{R}^n$ be a triangulable set and let $r$ be either a positive integer or $r=\infty$. We say that $Y$ is a $\scr{C}^r$-approximation target space, or a $\scr{C}^r$-${\tt ats}$ for short, if it has the following universal approximation property: {\it For each $m\in\Bbb{N}$ and each locally compact subset $X$ of $\Bbb{R}^m$, each continuous map $f:X\to Y$ can be approximated by $\scr{C}^r$ maps $g:X\to Y$ with respect to the strong Whitney $\scr{C}^0$ topology}. Taking advantage of new approximation techniques we prove: {\it if $Y$ is weakly $\scr{C}^r$ triangulable, then $Y$ is a $\scr{C}^r$-${\tt ats}$}. This result applies to relevant classes of triangulable sets, namely: (1) every locally compact polyhedron is a $\scr{C}^\infty$-${\tt ats}$, (2) every set that is locally $\scr{C}^r$ equivalent to a polyhedron is a $\scr{C}^r$-${\tt ats}$ (this includes $\scr{C}^r$ submanifolds with corners of $\Bbb{R}^n$) and (3) every locally compact locally definable set of an arbitrary o-minimal structure is a $\scr{C}^1$-${\tt ats}$ (this includes locally compact locally semialgebraic sets and locally compact subanalytic sets). In addition, we prove: {\it if $Y$ is a global analytic set, then each proper continuous map $f:X\to Y$ can be approximated by proper $\scr{C}^\infty$ maps $g:X\to Y$}. Explicit examples show the sharpness of our results.