# Past Analytic Topology in Mathematics and Computer Science

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.