  Metric topological vector spaces of Carath\'eodory functions and topologies
of $L^p_{loc}$ type are introduced, depending on a suitable set of moduli of
continuity. Theorems of continuous dependence on initial data for the solutions
of non-autonomous Carath\'eodory differential equations are proved in such new
topological structures. As a consequence, new families of continuous linearized
skew-product semiflows are provided in the Carath\'eodory spaces.
