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