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Without the restriction of metrizability, topological dynamical systems $(X,⟨T_s{⟩}_{s\in G})$ are defined and uniform recurrence and proximality are studied. Some well known results are generalized and some new results are obtained. In particular, a topological dynamical characterization of central sets in an arbitrary semigroup (G,+) is given and shown to be equivalent to the usual algebraic characterization.