Smooth Approximations in PL Geometry

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