Subword Metrics for Infinite Words

The space of one-sided infinite words plays a crucial rôle in several parts of Theoretical Computer Science. Usually, it is convenient to regard this space as a metric space, the Cantor-space. It turned out that for several purposes topologies other than the one of the Cantor-space are useful, e.g. for studying fragments of first-order logic over infinite words or for a topological characterisation of random infinite words. Continuing the work of [SS10], here we consider two different refinements of the Cantor-space, given by measuring common factors, and common factors occurring infinitely often. In particular we investigate the relation of these topologies to the sets of infinite words definable by finite automata, that is, to regular ω-languages