Abstract:Let $T$ be a permutation of an abstract set $X$. In ZFC, we find a necessary and sufficient condition it terms of cardinalities of the $T$-orbits that allows us to topologize $(X,T)$ as a topological dynamical system on a compact Hausdorff space. This extends an early result of H. de Vries concerning compact metric dynamical systems. An analogous result is obtained for ${\bold Z}^2$-actions without periodic points.
Keywords: abstract dynamical system, pointwise periodic system, symbolic dynamics, ${\bold Z}^2$-action
AMS Subject Classification: 54H20