  We prove that every real algebraic set $V\subset\mathbb{R}^n$ with isolated
singularities is homeomorphic to a set $V'\subset\mathbb{R}^m$ that is
$\mathbb{Q}$-algebraic in the sense that $V'$ is defined in $\mathbb{R}^m$ by
polynomial equations with rational coefficients. The homeomorphism $\phi:V\to
V'$ we construct is semialgebraic, preserves nonsingular points and restricts
to a Nash diffeomorphism between the nonsingular loci. In addition, we can
assume that $V'$ has a codimension one subset of rational points. If $m$ is
sufficiently large, we can also assume that $V'\subset\mathbb{R}^m$ is
arbitrarily close to $V\subset\mathbb{R}^n\subset\mathbb{R}^m$, and $\phi$
extends to a semialgebraic homeomorphism from $\mathbb{R}^m$ to $\mathbb{R}^m$.
A first consequence of this result is a $\mathbb{Q}$-version of the
Nash-Tognoli theorem: Every compact smooth manifold admits a
$\mathbb{Q}$-algebraic model. Another consequence concerns the open problem of
making Nash germs $\mathbb{Q}$-algebraic: Every Nash set germ with an isolated
singularity is semialgebraically equivalent to a $\mathbb{Q}$-algebraic set
germ.
