Joint Topologies for Finite and Infinite Words

Abstract

Infinite words are often considered as limits of finite words. As topological methods have been proved to be useful in the theory of ω-languages it seems to be providing to include finite and infinite words into one (topological) space. The attempts so far (see [3, Section 2.4]) have their drawbacks.

Therefore, in the present paper we investigate the possibility to join separate topologies on the space of finite words with a topology in the space of infinite words via a natural mapping. A requirement in this linking of topologies consists in the compatibility of topological properties (opennenss, closedness etc) of images with pre-images and vice versa.

Here we choose the natural Cantor topology for infinite words and the δ-limit as linking mapping, nad we show that several natural topologies on the space of finite words prove to be compatible with the topology of the Cantor space. It is interesting to observe that besides the well-known prefix topology there are at least two more whose origin is from language theory, from the construction of centers and super-centers of languages.

These center- and supercenter-topologies on the space of finite words, $\mathcal{T}_{c}$ and $\mathcal{T}_{sc}$, respectively, fit into the class of $\mathcal{L}$-topologies investigated in [2]. Moreover they exhibit special properties within the classes of topologies compatible with the Cantor topology.

The paper presents the main results, omitting, however, due to lack of space the results on $\mathcal{L}$-topologies. For notation see Sections 1 and 2.1 of [3], and for topological background see e.g. [1].