Arguments on PL (=piecewise linear) topology work over any ordered field in the same way as over the real number field, and those on differential topology do over a real closed field in an o- minimal structure that expands (R,<,0,1,+,\cdot). It is known that a compact definable set is definably homeomorphic to a polyhedron.We show uniqueness of the polyhedron up to PL homeomorphism (o-minimal Hauptvermutung).We see also that many problems on PL and differential topology can be translated to those over the real number field.