Algebraic approximation preserving dimension
  We prove that each semialgebraic subset of $\R^n$ of positive codimension can be locally approximated of any order by means of an algebraic set of the same dimension. As a consequence of previous results, algebraic approximation preserving dimension holds also for semianalytic sets. 
Massimo Ferrarotti, Elisabetta Fortuna, Leslie Wilson
math.AG