Topology on words

Abstract

Order-based topologies a b s t r a c tWe investigate properties of topologies on sets of finite and infinite words over a finite alphabet. The guiding example is the topology generated by the prefix relation on the set of finite words, considered as a partial order. This partial order extends naturally to the set of infinite words; hence it generates a topology on the union of the sets of finite and infinite words. We consider several partial orders which have similar properties and identify general principles according to which the transition from finite to infinite words is natural. We provide a uniform topological framework for the set of finite and infinite words to handle limits in a general fashion.

Fragmentary Fuzzy Pseudometrics: Basics of the Theory and Applications in Combinatorics on Words

“…Similar to metric ρ, is the so called Cantor metric c : X × X → [0, 1], see e.g. (Calude et al, 2009) which also fully depends on the first part, that is on the prefix of the word.…”

“…The paper (Calude et al, 2009) contains a survey of different attempts to introduce an appropriate topology on such sets. The authors distinguish two different approaches to this problem.…”

“…However, since any natural order on this set is heavily dependent on the starting part, that is the prefix of the word, the resulting topology has a also an typical prefix character. The authors of (Calude et al, 2009) describe different principles of defining meaningful topologies on the set of infinite words and conclude that although some of these principles are not fully restricted to the prefix order of words itself, but still they rely on it quite heavily. In conclusion the authors say "It should be possible to derive far more general principles which apply to many more relations between words by changing the intuition about words being read left to right. "…”

Topologies Refining the Cantor Topology on X ω

“…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].…”

“…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].…”