TOPOLOGIES FOR THE SET OF DISJUNCTIVE ω-WORDS
Ludwig Staiger · WORLD SCIENTIFIC eBooks · 2001
An infinite sequence (!-word) is referred to as disjunctive provided it contains every finite word as infix (factor). As Jrgensen and Thierrin [JT83] observed the set of disjunctive !-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 !-words. We introduce two non-standard topologies on the set of all !-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 G -set, and in the second one D is the set of accumulation points of the whole space as well as of itself.