Equivariant semialgebraic vector bundles

Let G be a compact semialgebraic group. We prove that any semialgebraic G-vector bundle over a semialgebraic G-set has a semialgebraic classifying G-map. Moreover we prove that the set of semialgebraic G-isomorphism classes of semialgebraic G-vector bundles over a semialgebraic G-set corresponds bijectively to the set of topological G-isomorphism classes of topological G-vector bundles over it. As an application of them, we prove that for any two semialgebraic G-maps f,h between semialgebraic G-sets M and N, if they are G-homotopic and ξ is a semialgebraic G-vector bundle over N, then f∗(ξ) and h∗(ξ) are semialgebraically G-isomorphic.