This talk is about a subject about which a great deal is known: the existence, uniqueness, and regularity of solutions of ordinary differential equations. The differential equations of interest are time-varying with very irregular (measurable) time-dependence and very regular (analytic) dependence on state. For such equations, one wants a theory ensuring that dependence on initial conditions is as regular as that of the differential equation on state. It turns out that it is not immediately obvious how to formulate the correct conditions on the right-hand side to achieve the desired regularity in initial condition. In this work, the correct conditions are formulated by topologising the set of right-hand sides and then formulating conditions in terms of this topology. As a cute byproduct, this topological machinery gives a pithy description of the classical existence, uniqueness, and regularity conditions.
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