Abstract:
Infinite words are often considered as limits of finite words. As
topological methods have been proved to be useful in the theory of
!-languages it seems to be providing to include finite and infinite
words into one (topological) space. In most cases this results in a
poor topological structure induced on the subspace of finite words.
In the present paper we investigate the possibility to link topologies
in 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 the topological properties
(openness, closedness etc) of images with preimages and vice
versa.
Here we show that choosing for infinite words the natural topology
of the CANTOR space and the -limit as linking mapping there
are several natural topologies on the space of finite words 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 fromlanguage theory—centers and supercenters
of languages.
We show that several of these topologies on the space of finite
words fit into a class of L-topologies and exhibit their special properties
w.r.t. to the compatibility with the CANTOR topology.