Topologies for the set of disjunctive ω-words

Authors:
Ludwig Staiger
Affiliations:
Institut für Informatik, Martin-Luther-Universität Halle-Wittenberg, Halle, Germany
Venue:
Acta Cybernetica
Year:
2005

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Abstract

An infinite sequence (ω-word) is referred to as disjunctive provided it contains every finite word as infix (factor). As Jürgensen and Thierrin [JT83] observed the set of disjunctive ω-words, D, has a trivial syntactic monoid but is not accepted by a finite automaton.In this paper we derive some topological properties of the set of disjunctive ω-words. We introduce two non-standard topologies on the set of all ω-words and show that D fulfills some special properties with respect to these topologies:In the first topology - the so-called topology of forbidden words - D is the smallest nonempty Gδ-set, and in the second one D is the set of accumulation points of the whole space as well as of itself.