Based on the theory of Carathéodory structure, this paper introduces the topological entropy of a flow on non-compact sets. Moreover, we introduce the definition of measure-theoretic entropy of a flow. It is shown that this entropy is equivalent to the one defined by Sun in [10]. The variational principle between topological entropy and measure-theoretic entropy of a flow is established. We also get the Brin-Katok's entropy formula for a flow.
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