Original manuscript: 2023/11/19

A framework is presented for ordinary differential equations with measurable time-dependence and with parameter-dependence in a general topological space. This framework has both geometric and function analytic aspects. The geometric aspect is reflected by the framework being that of vector fields and flows on manifolds. The function analytic aspect is reflected by the classes of vector fields and flows being characterised by function space topologies. These classes of vector fields and mappings are presented across a variety of regularity classes which includes Lipschitz, finitely differentiable, smooth, real analytic, and holomorphic. Special emphasis is placed on the rôle of composition operators, particularly in time- and parameter-dependent settings.

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