Topologies Refining the Cantor Topology on X ω

Abstract

Abstract. 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.It is shown that both of 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 respect, the definability of sets of infinite words by finite automata.The space of one-sided infinite words plays a crucial rôle in several parts of Theoretical Computer Science (see [7,20] and the surveys [6,13,17,18]). Usually, it is convenient to regard this space as a topological space provided with the Cantor topology. This topology can be also considered as the natural continuation of the left topology of the prefix relation on the space of finite words; for a survey see [2].It turned out that for several purposes other topologies on the space of infinite words are also useful [9,12], e.g. for investigations in first-order logic [3], to characterise the set of random infinite words [1] or the set of disjunctive infinite words [15] and to describe the converging behaviour of not necessarily hyperbolic iterative function systems [5,14].Most of these papers use topologies on the space of infinite words which are certain refinements of the Cantor topology showing a certain kind of shift invariance. The aim of this paper is to give a unified treatment of those topologies and to investigate their relations to Cantor topology.Special attention is paid to subsets of the space of infinite words definable by finite automata. It turns out that several of the refinements of the Cantor topology under consideration behave well with respect to finite automata, that is, the corresponding closure and interior operators preserve at least one of the classes of finite-state or regular ω-languages.

Level Two of the Quantifier Alternation Hierarchy Over Infinite Words

“…The alphabetic topology has by itself been the subject of further study [15]. We are particularly interested in Boolean combinations of open sets.…”

Level Two of the Quantifier Alternation Hierarchy over Infinite Words

“…The alphabetic topology has been introduced in [4], where it is used as a part of the characterization of Σ 2 over Γ ∞ . The alphabetic topology has by itself been the subject of further study [15]. We are particularly interested in Boolean combinations of open sets.…”