Analytic Spaces and their Tukey Types
Advisor:
Todorcevic, Stevo
Department:
Mathematics
Issue Date:
Nov-2019
Abstract (summary):
In this Thesis, we study topologies on countable sets from the perspective of Tukey reductions
of their neighbourhood filters. It turns out that is closely related to the already established
theory of definable (and in particular analytic) topologies on countable sets. The connection
is, in fact, natural as the neighbourhood filters of points in such spaces are typical examples of
directed sets for which Tukey theory was introduced some eighty years ago. What is interesting
here is that the abstract Tukey reduction of a neighbourhood filter Fx of a point to standard
directed sets like or ℓ1 imposes that Fx must be analytic. We develop a theory that examines
the Tukey types of analytic topologies and compare it with the theory of sequential convergence
in arbitrary countable topological spaces either using forcing extensions or axioms such as, for
example, the Open Graph Axiom. It turns out that in certain classes of countable analytic
groups we can classify all possible Tukey types of the corresponding neighbourhood filters of
identities. For example, we show that if G is a countable analytic k-group then 1 = {0}, and
are the only possible Tukey types of the neighbourhood filter F
G
e
. This will give us also
new metrization criteria for such groups. We also show that the study of definable topologies
on countable index sets has natural analogues in the study of arbitrary topologies on countable
sets in certain forcing extensions.
Permanent Link:
https://hdl.handle.net/1807/97474
Content Type:
Thesis
Items in TSpace are protected by copyright, with all rights reserved, unless otherwise indicated.
link to htmlmap