In this article, we show that a flat morphism of $k$-varieties ($\mathop{ \mathrm{ char}} k=0$) with locally constant geometric fibers becomes finite \'etale after reduction. When $k$ is a real closed field, we prove that such a morphism induces a covering map on the rational points. We further give a triviality result different from Hardt's and a new interpretation of the construction of cylindrical algebraic decomposition as applications.
Categories: math.AG cs.SC