Shift-invariant topologies for the Cantor space Xω

It is shown that these topologies refine the topology of the Cantor space. Moreover, from common features of these topologies we extract properties which characterise a large class of topologies. It turns out that, for this general class of topologies, the corresponding closure and interior operators respect the shift operations and also, to some extent, the definability of sets of infinite words by finite automata.