In the 1980's H. Delfs and M. Knebusch developed a homotopy theory for locally semialgebraic and weakly semialgebraic spaces (see  and ).
Such spaces may be obtained by glueing definable sets of real closed fields along open, or properly chosen closed definable subsets. Then we can do semialgebraic homotopy theory by imitating topological homotopy theory but restricting our attention to the sets and mappings from the appropriate category.
It appears that under some natural assumptions the absolute or relative homotopy classes of systems of spaces are stable under the base field extension and that they are equal to usual topological homotopy classes if the spaces are over the field of real numbers.
This applies in particular to the absolute or relative homotopy groups and has further consequences.
The theory is extended in the following way:
We assume that R is an o-minimal expansion of a real closed field. We consider similarly defined locally and weakly definable spaces over R, and get analogous theorems
 H. Delfs, M. Knebusch, Locally Semialgebraic Spaces, Lecture Notes in Mathematics 1173, Springer-Verlag 1985.
 M. Knebusch, Weakly Semialgebraic Spaces, Lecture Notes in Mathematics 1367, Springer-Verlag 1989.