Past Analytic Topology in Mathematics and Computer Science

It is known that every continuum X is a weakly confluent image of a continuum Y which is hereditarily indecomposable and of covering dimension one.  We use the ultracoproduct construction to gain information about the number of composants of Y.  For example, in ZFC, we can ensure that this number is arbitrarily large.  And if we assume the GCH, we can arrange for Y to have as many composants at it has points.

Abstract:   Let $f$ be a continuous surjection from the compact metric space $X$ to itself. 

We say that the dynamical system $(X,f)$ has shadowing if for every $\epsilon>0$ there is a $\delta>0$ such that every $\delta$-pseudo orbit is $\epsilon$-shadowed.  Here a sequence $(x_n)$ is a $\delta$-pseudo orbit provided the distance from $f(x_n)$ to $x_{n+1}$ is less than $\delta$ and $(x_n)$ is $\epsilon$-shadowed if there is a point $z$ such that the distance from $x_n$ to $f^n(z)$ is less than $\epsilon$.  

If $f$ is a homeomorphism, $(X,f)$ is said to be expansive if there is some $c>0$, such that if the distance from $f^n(x)$ and $f^n(y)$ is less than $c$ for all $n\in \mathbb Z$, then $x=y$.

In his proof that a homeomorphism that is expansive and has shadowing is stable, Walters shows that in an expansive system with shadowing, a pseudo orbit is shadowed by exactly one point.  It turns out that the converse is also true: if the system has unique shadowing (in the above sense), then it is expansive.

In this talk, which is joint work with Joel Mitchell and Joe Thomas, we explore this notion of unique shadowing.

Abstract:  It is a central question in the theory of infinite relational structures as to which structures carry analogues of Ramsey’s Theorem.  This question, of interest for several decades, has gained recent momentum as it was brought into focus by Kechris, Pestov, and Todorcevic, when they proved a deep correspondence between Ramsey theory and topological dynamics.  

In this talk, we provide background on the Ramsey theory of the Rado graph, solved by Sauer.  A longstanding open question was whether Henson graphs, the k-clique-free analogues of the Rado graph, have similar features.  We present the speaker’s recent work solving the Ramsey theory of the Henson graphs.  The techniques developed open new lines of investigation for other relational structures with forbidden configurations.  As a byproduct of these methods, we may obtain Ramsey properties for Borel colorings on copies of the Rado graph, with respect to a certain topology.

Abstract:  It is a central question in the theory of infinite relational structures as to which structures carry analogues of Ramsey’s Theorem.  This question, of interest for several decades, has gained recent momentum as it was brought into focus by Kechris, Pestov, and Todorcevic, when they proved a deep correspondence between Ramsey theory and topological dynamics.  

In this talk, we provide background on the Ramsey theory of the Rado graph, solved by Sauer.  A longstanding open question was whether Henson graphs, the k-clique-free analogues of the Rado graph, have similar features.  We present the speaker’s recent work solving the Ramsey theory of the Henson graphs.  The techniques developed open new lines of investigation for other relational structures with forbidden configurations.  As a byproduct of these methods, we may obtain Ramsey properties for Borel colorings on copies of the Rado graph, with respect to a certain topology.

Every topological space is metrisable once the symmetry axiom is abandoned and the codomain of the metric is allowed to take values in a suitable structure tailored to fit the topology (and every completely regular space is similarly metrisable while retaining symmetry). This result was popularised in 1988 by Kopperman, who used value semigroups as the codomain for the metric, and restated in 1997 by Flagg, using value quantales. In categorical terms, each of these constructions extends to an equivalence of categories between the category Top and a category of all L-valued metric spaces (where L ranges over either value semigroups or value quantales) and the classical \epsilon-\delta notion of continuous mappings. Thus, there are (at least) two metric formalisms for topology, raising the questions: 1) is any of the two actually useful for doing topology? and 2) are the two formalisms equally powerful for the purposes of topology? After reviewing Flagg's machinery I will attempt to answer the former affirmatively and the latter negatively. In more detail, the two approaches are equipotent when it comes to point-to-point topological consideration, but only Flagg's formalism captures 'higher order' topological aspects correctly, however at a price; there is no notion of product of value quantales. En route to establishing Flagg's formalism as convenient, it will be shown that both fine and coarse variants of homology and homotopy arise as left and right Kan extensions of genuinely metrically constructed functors, and a topologically relevant notion of tensor product of value quantales, a surrogate for the non-existent products, will be described. 


 

Nash-Williams showed that the collection of locally finite trees under the topological minor relation results in a BQO. Naturally, two interesting questions arise:

1.      What is the number \lambda of topological types of locally finite trees?

2.       What are the possible sizes of an equivalence class of locally finite trees?

 For (1), clearly, \omega_0 \leq \lambda \leq c and Matthiesen refined it to \omega_1 \leq \lambda \leq c. Thus, this question becomes non-trivial in the absence of the Continuum Hypothesis. In this paper we address both questions by showing - entirely within ZFC - that for a large collection of locally finite trees that includes those with countably many rays:

- \lambda = \omega_1, and

- the size of an equivalence class can only be either 1 or c.

A main part of the proof uses forcing to establish a Ramsey theorem on a new type of tree, though the result holds in ZFC.  The space of such trees almost forms a topological Ramsey space.

The talk will focus on five items:

Theorem 1. It is ZFC-independent whether every locally compact, $\omega_1$-compact space of cardinality $\aleph_1$  is the union of countably many countably compact spaces.

Problem 1. Is it consistent that every locally compact, $\omega_1$-compact space of cardinality $\aleph_2$  is the union of countably many countably compact spaces?

[`$\omega_1$-compact' means that every closed discrete subspace is countable. This is obviously implied by being the union of countably many countably compact spaces, but the converse is not true.]

Problem 2. Is ZFC enough to imply that there is  a normal, locally countable, countably compact space of cardinality greater than $\aleph_1$?

Problem 3. Is it consistent that there exists a normal, locally countable, countably compact space of cardinality greater than $\aleph_2$?

The spaces involved in Problem 2 and Problem 3 are automatically locally compact, because by "space" I mean "Hausdorff space" and so regularity is already enough to give every point a countable countably compact (hence compact) neighborhood.

Theorem 2. The axiom $\square_{\aleph_1}$ implies that there is a normal, locally countable, countably compact space of cardinality $\aleph_2$.

This may be the first application of $\square_{\aleph_1}$ to construct a topological space whose existence in ZFC is unknown.

Abstract:  Anyone who has worked in $\beta $N will not be surprised to learn that some of the algebraically defined subsets of $\beta N$ are not topologically simple, even though their algebraic definition may be very simple.  I shall show that the following subsets of $\beta N$ are not Borel: $N^*+N^*$; the smallest ideal of $\beta N$; the set of idempotents in $\beta N$; any semiprincipal right ideal in $\beta N$; the set of idempotents in any left ideal in $\beta N$.

 Anyone who has worked in $\beta $N will not be surprised to learn that some of the algebraically defined subsets of $\beta N$ are not topologically simple, even though their algebraic definition may be very simple.  I shall show that the following subsets of $\beta N$ are not Borel: $N^*+N^*$; the smallest ideal of $\beta N$; the set of idempotents in $\beta N$; any semiprincipal right ideal in $\beta N$; the set of idempotents in any left ideal in $\beta N$.

Partial metric spaces generalise metric spaces by allowing self-distance
to be a non-negative number. Originally motivated by the goal to
reconcile metric space topology with the logic of computable functions
and Dana Scott's innovative theory of topological domains they are now
too rigid a form of mathematics to be of use in modelling contemporary
applications software (aka 'Apps') which is increasingly concurrent,
pragmatic, interactive, rapidly changing, and inconsistent in nature.
This talks aims to further develop partial metric spaces in order to
catch up with the modern computer science of 'Apps'. Our illustrative
working example is that of the 'Lucid' programming language,and it's
temporal generalisation using Wadge's 'hiaton'.

ABSTRACT: A space of countable extent, also called an omega_1-compact space, is one in which every closed discrete subspace is countable.  The axiom used in the following theorem is consistent if it is consistent that there is a supercompact cardinal.

Theorem 1  The LCT axiom implies that every hereditarily normal, omega_1-compact space
is sigma-countably compact,  i.e., the union of countably many countably compact subspaces.

Even for the specialized subclass of monotonically normal spaces, this is only a consistency result:

Theorem 2   If club, then there exists a locally compact, omega_1-compact monotonically
normal space that is not sigma-countably compact.

These two results together are unusual in that most independence results on
monotonically normal spaces depend on whether Souslin's Hypothesis (SH) is true,
and do not involve large cardinal axioms. Here, it is not known whether either
SH or its negation affect either direction in this independence result.

The following unsolved problem is also discussed:

Problem  Is there a ZFC example of a locally compact, omega_1-compact space
of cardinality aleph_1 that is not sigma-countably compact?

It is quite easy to see that the sobrification of a
topological space is a dcpo with respect to its specialisation order
and that the topology is contained in the Scott topology wrt this
order. It is also known that many classes of dcpo's are sober when
considered as topological spaces via their Scott topology. In 1982,
Peter Johnstone showed that, however, not every dcpo has this
property in a delightful short note entitled "Scott is not always
sober".

Weng Kin Ho and Dongsheng Zhao observed in the early 2000s that the
Scott topology of the sobrification of a dcpo is typically different
from the Scott topology of the original dcpo, and they wondered
whether there is a way to recover the original dcpo from its
sobrification. They showed that for large classes of dcpos this is
possible but were not able to establish it for all of them. The
question became known as the Ho-Zhao Problem. In a recent
collaboration, Ho, Xiaoyong Xi, and I were able to construct a
counterexample.

In this talk I want to present the positive results that we have about
the Ho-Zhao problem as well as our counterexample. 

"In a paper from 2001, Diestel and Leader characterised uncountable graphs with normal spanning trees through a class of forbidden minors. In this talk we investigate under which circumstances this class of forbidden minors can be made nice. In particular, we will see that there is a nice solution to this problem under Martin’s Axiom. Also, some connections to the Stone-Chech remainder of the integers, and almost disjoint families are uncovered.”

Point-free topology can often seem like an algebraic almost-topology, 
> not quite the same but still interesting to those with an interest in 
> it. There is also a tradition of it in computer science, traceable back 
> to Scott's topological model of the untyped lambda-calculus, and 
> developing through Abramsky's 1987 thesis. There the point-free approach 
> can be seen as giving new insights (from a logic of observations), 
> albeit in a context where it is equivalent to point-set topology. It was 
> in that tradition that I wrote my own book "Topology via Logic".
> 
> Absent from my book, however, was a rather deeper connection with logic 
> that was already known from topos theory: if one respects certain 
> logical constraints (of geometric logic), then the maps one constructs 
> are automatically continuous. Given a generic point x of X, if one 
> geometrically constructs a point of Y, then one has constructed a 
> continuous map from X to Y. This is in fact a point-free result, even 
> though it unashamedly uses points.
> 
> This "continuity via logic", continuity as geometricity, takes one 
> rather further than simple continuity of maps. Sheaves and bundles can 
> be understood as continuous set-valued or space-valued maps, and topos 
> theory makes this meaningful - with the proviso that, to make it run 
> cleanly, all spaces have to be point-free. In the resulting fibrewise 
> topology via logic, every geometric construction of spaces (example: 
> point-free hyperspaces, or powerlocales) leads automatically to a 
> fibrewise construction on bundles.
> 
> I shall present an overview of this framework, as well as touching on 
> recent work using Joyal's Arithmetic Universes. This bears on the logic 
> itself, and aims to replace the geometric logic, with its infinitary 
> disjunctions, by a finitary "arithmetic type theory" that still has the 
> intrinsic continuity, and is strong enough to encompass significant 
> amounts of real analysis.

We know that the existence of a period three point for an interval map implies much about the dynamics of the map, but the restriction of the map to the periodic orbit itself is trivial. Countable invariant subsets arise naturally in many dynamical systems, for example as $\omega$-limit sets, but many of the usual notions of dynamics degenerate when restricted to countable sets. In this talk we look at what we can say about dynamics on countable compact spaces.  In particular, the theory of countable dynamical systems is the theory of the induced dynamics on countable invariant subsets of the interval and the theory of homeomorphic countable dynamics is the theory of compact countable invariant subsets of homeomorphisms of the plane.

Joint work with Columba Perez

CANCELLED - CANCELLED - CANCELLED

Abstract: Joint work with Syahida Che Dzul-Kifli

Let $f:X\to X$ be a continuous function on a compact metric space forming a discrete dynamical system. There are many definitions that try to capture what it means for the function $f$ to be chaotic. Devaney’s definition, perhaps the most frequently cited, asks for the function $f$ to be topologically transitive, have a dense set of periodic points and is sensitive to initial conditions.  Bank’s et al show that sensitive dependence follows from the other two conditions and Velleman and Berglund show that a transitive interval map has a dense set of periodic points.  Li and Yorke (who coined the term chaos) show that for interval maps, period three implies chaos, i.e. that the existence of a period three point (indeed of any point with period having an odd factor) is chaotic in the sense that it has an uncountable scrambled set.

The existence of a period three point is In this talk we examine the relationship between transitivity and dense periodic points and look for simple conditions that imply chaos in interval maps. Our results are entirely elementary, calling on little more than the intermediate value theorem.

Compact F-spaces play an important role in the area of compactification theory, the prototype being w*, the Stone-Cech remainder of the integers. Two curious topological characteristics of compact F-spaces are that they don’t contain convergent sequences (apart from the constant ones), and moreover, that they often contain points that don’t lie in the boundary of any countable subset (so-called weak P-points). In this talk we investigate the space of self-maps S(X) on compact zero-dimensional F-spaces X, endowed with the compact-open topology. A natural question is whether S(X) reflects properties of the ground space X. Our main result is that for zero-dimensional compact F-spaces X, also S(X) doesn’t contain convergent sequences. If time permits, I will also comment on the existence of weak P-points in S(X). This is joint work with Richard Lupton.