Institute of Informatics
Acta Cybernetica
Past Issues
Volume 17, Number 1, 2005
Topologies for the Set of Disjunctive $\omega$-words
# Topologies for the Set of Disjunctive $\omega$-words

**Ludwig Staiger**

### Abstract (in LaTeX format)

An infinite sequence ($\omega$-word) is referred to as disjunctive provided it contains every finite word as infix (factor). As J\"urgensen and Thierrin observed the set of disjunctive $\omega$-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 $\omega$-words. We introduce two non-standard topologies on the set of all $\omega$-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 $\mathbf{G}_\delta$-set, and in the second one $D$ is the set of accumulation points of the whole space as well as of itself.

### Full text

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### DOI

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### BibTeX entry
`
@article{Staiger:2005:ActaCybernetica,`

author = {Ludwig Staiger},

title = {Topologies for the Set of Disjunctive $\omega$-words},

journal = {Acta Cybernetica},

year = {2005},

volume = {17},

number = {1},

pages = {43--51},

abstract = {An infinite sequence ($\omega$-word) is referred to as disjunctive provided it contains every finite word as infix (factor). As J\"urgensen and Thierrin observed the set of disjunctive $\omega$-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 $\omega$-words. We introduce two non-standard topologies on the set of all $\omega$-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 $\mathbf{G}_\delta$-set, and in the second one $D$ is the set of accumulation points of the whole space as well as of itself.}

}