  We introduce new weak topologies and spaces of Carath\'eodory functions where
the solutions of the ordinary differential equations depend continuously on the
initial data and vector fields. The induced local skew-product flow is proved
to be continuous, and a notion of linearized skew-product flow is provided. Two
applications are shown. First, the propagation of the exponential dichotomy
over the trajectories of the linearized skew-product flow and the structure of
the dichotomy or Sacker-Sell spectrum. Second, how particular bounded absorbing
sets for the process defined by a Carath\'eodory vector field $f$ provide
bounded pullback attractors for the processes with vector fields in the
alpha-limit set, the omega-limit set or the whole hull of $f$. Conditions for
the existence of a pullback or a global attractor for the skew-product
semiflow, as well as application examples are also given.
