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.

# Past Analytic Topology in Mathematics and Computer Science

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