# Article

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Keywords:
abstract dynamical system; pointwise periodic system; symbolic dynamics; \$\bold Z^2\$-action
Summary:
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.
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