Convex real projective structures on closed surfaces are closed

The deformation space $\mathfrak{C}(\Sigma )$ of convex $\mathbb{R}{{\mathbf{P}}^2}$-structures on a closed surface $\Sigma$ with $\chi (\Sigma ) < 0$ is closed in the space $\operatorname{Hom} (\pi ,\operatorname{SL} (3,\mathbb{R}))/\operatorname{SL} (3,\mathbb{R})$ of equivalence classes of representations ${\pi _1}(\Sigma ) \to \operatorname{SL} (3,\mathbb{R})$. Using this fact, we prove Hitchin's conjecture that the contractible "Teichmüller component" (Lie groups and Teichmüller space, preprint) of $\operatorname{Hom} (\pi ,\operatorname{SL} (3,\mathbb{R}))/\operatorname{SL} (3,\mathbb{R})$ precisely equals $\mathfrak{C}(\Sigma )$.