This paper considers dynamical systems from a topological point of view. The authors extend various notions and generalize many results in the theory of dynamical systems from the setting of metric spaces to the setting of Hausdorff spaces. To achieve this purpose, the authors use two approaches: one is in terms of finite open covers and the other is in terms of uniformities (compatible with the topology). In the presence of compactness, where there is a unique compatible uniformity, these two approaches are equivalent and in compact metric spaces they coincide exactly with the standard definition. Among many other things, the authors prove that in a Tychonoff space, transitivity and dense periodic points imply (uniform) sensitivity to initial conditions, and generalize \textit{B. F. Bryant}'s classical result [Pac. J. Math. 10, 1163--1167 (1960; Zbl 0101.15504)] that a compact Hausdorff space admitting a \(c\)-expansive homeomorphism in the obvious uniform sense is metrizable. Different versions of shadowing and internal chain transitivity are also investigated.