On non-separating simple closed curves in a compact surface

We introduce a semi-algebraic structure on the set $\mathcal{S}$ of all isotopy classes of non-separating simple closed curves in any compact oriented surface and show that the structure is finitely generated. As a consequence, we produce a natural finite dimensional linear representation of the mapping class group of the surface. Applications to the Teichmüller space, Thurston's measured lamination space, the harmonic Beltrami differentials, and the first cohomology group of the surface are discussed.