Let $T\subset{ \mathbb R}^n$ be a semialgebraic set and let $\mu\ge0$ be a non-negative integer. We say that $T$ is a { \em Nash $\mu$-approximation target space} (or a $({ \mathcal N},\mu)$-${ \tt ats}$ for short) if it has the following universal approximation property: { \em For each $m\in{ \mathbb N}$ and each locally compact semialgebraic subset $S\subset{ \mathbb R}^m$, the subspace of Nash maps ${ \mathcal N}(S,T)$ is dense in the space ${ \mathcal S}^\mu(S,T)$ of ${ \mathcal C}^\mu$ semialgebraic maps between $S$ and $T$}. A necessary condition to be a $({ \mathcal N},\mu)$-${ \tt ats}$ is that $T$ is locally connected by analytic paths. In this paper we show: { \em Nash manifolds with corners are $({ \mathcal N},\mu)$-${ \tt ats}$ for each $\mu\geq0$}. As an application of a stronger version of the previous statement, we show that if two Nash maps $f,g:S\to Q$, where $S$ is a locally compact semialgebraic set of ${ \mathbb R}^m$ and $Q$ is a Nash manifold with corners, are close enough in the (strong) Whitney's semialgebraic topology of ${ \mathcal S}^0(S,T)$ (and consequently they are (continuous) semialgebraically homotopic), then $f,g$ are Nash homotopic.
Categories: math.AG