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 $\beta N$.

# Past Analytic Topology in Mathematics and Computer Science

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 $\beta N$.

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?