T. Mostowski showed that every (real or complex) germ of an analytic set is homeomorphic to the germ of an algebraic set. In this paper we show that every (real or complex) analytic function germ, defined on a possibly singular analytic space, is topologically equivalent to a polynomial function germ defined on an affine algebraic variety.

Let X be a compact nonsingular affine real algebraic variety. We prove that every pre-algebraic vector bundle on X becomes algebraic after finitely many blowing ups. Using this theorem, we then prove that the Stiefel-Whitney classes of any pre-algebraic R-vector bundle on X are algebraic. We also derive that the Chern classes of any pre-algebraic C-vector bundles and the Pontryagin classes of any pre-algebraic R-vector bundle are blow-C-algebraic. We also provide several results on line bundles on X .

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