Bierstone and Parusiński studied the desingularization of $d$-dimensional closed subanalytic sets and in particular of $d$-dimensional closed semialgebraic sets. Their procedure preserves the number of $d$-dimensional components connected by analytic paths of the involved closed semialgebraic sets, so they have a good behaviour for pure dimensional closed semialgebraic sets. If the involved $d$-dimensional closed semialgebraic set is not pure dimensional, some components connected by analytic paths of smaller dimension could be dropped during the desingularization process. For instance, Whitney's umbrella $W:=\{{\tt y}^2{\tt z}-{\tt x}^2=0\}\subset\mathbb{R}^3$ has two components connected by analytic paths (one of dimension $2$ and the other of dimension $1$), whereas its desingularization has only one component connected by analytic paths, that has dimension $2$. The obtained models in the desingularization process, that we call closed chessboard sets, are the closures of (finite) unions of connected components of the complements of normal-crossings divisors of non-singular real algebraic sets.

In this seminar we start by presenting the resolution of $d$-dimensional closed chessboard sets $\mathcal{S}$ using Nash manifolds with corners $\mathcal{Q}$ with the same number of connected components as $\mathcal{S}$ (or equivalently the same number of irreducible components). Next, we present the Nash double $D(\mathcal{Q})$ of a Nash manifold with corners $\mathcal{Q}$, which is the analogous of the Nash double of a Nash manifold with smooth boundary, but takes into account the peculiarities of the boundary of $\mathcal{Q}$. Combining the previous results with Bierstone and Parusiński's desingularization (together with some algebraization techniques) we obtain that: If $\mathcal{S}$ is a $d$-dimensional closed semialgebraic set connected by analytic paths, then there exists a $d$-dimennsional non-singular irreducible real algebraic set $X$, a proper surjective polynomial map $f:X\to \mathcal{S}$ and a closed semialgebraic subset $\mathcal{R}\subset \mathcal{S}$ of dimension $<d$ such that $X\setminus f^{-1}(\mathcal{R})$ and $\mathcal{S}\setminus \mathcal{R}$ are Nash manifolds of the same dimension $d$ and the restriction of $f$ to $X\setminus f^{-1}(\mathcal{R})$ is a Nash covering map whose fibers are finite and have constant cardinality. Finally, we present several applications of our results:

• A complete characterisation of the compact semialgebraic sets that are images of the unitary closed ball under Nash maps.
• A compactification result for Nash manifolds with corners by compact Nash manifolds with corners.
• A result on Nash approximation of continuous semialgebraic maps whose target space are Nash manifolds with corners.