Topological Domain Theory
Topological domain theory is
a generalization of domain theory
that includes a wider collection of topological spaces than
traditional domain theory. The generalization
overcomes certain known limitations of domain theory,
which is unable to model various awkward
combinations of computational features.
The development of topological domain theory was supported by
an EPSRC research grant "Topological Models for Computational
Metalanguages" 20036, which employed Matthias Schröder
and Ingo Battenfeld. Here is the final report for the
project (postscript,
pdf).
Overviews of topological domain theory
A manifesto for topological domain theory was presented in
talks at Dagstuhl and Kyoto, 2002.

Towards a Convenient Category of Topological Domains
Alex Simpson
In Proceedings of
Thirteenth ALGI Workshop, RIMS, Kyoto University, 2003
(postscript,
pdf)
An uptodate overview of research in topological domain theory
will appear in Gordon Plotkin's forthcoming Festschrift:

A Convenient Category of Domains
Ingo Battenfeld, Matthias Schröder and Alex Simpson
Computation, Meaning and Logic,
Articles dedicated to Gordon Plotkin
L. Cardelli, M. Fiore and G. Winskel (eds)
Electronic Notes in Computer Science, 34 pp., to appear, 2007
(postscript,
pdf)
Mathematical development of topological domain theory
Topological domain theory
is based on the remarkable closure properties of qcb spaces
(topological quotients of
countably based spaces).
These study of these spaces was initiated independently
by Schröder and by Menni and Simpson in the late 1990's.
Qcb spaces provide a link between general topology in mathematics,
realizability semantics in computer science, and
computability theory (especially, Weihrauch's
type two theory of effectivity).
Here is a selection of background material developing
these aspects of qcb spaces.

Topological and Limitspace Subcategories of
Countablybased Equilogical Spaces.
Matías Menni and Alex Simpson
Mathematical Structures in Computer Science, 12:739770, 2002
(postscript,
pdf)

Admissible Representations for Continuous Computations
Matthias Schröder
PhD Thesis, Fachbereich Informatik,
FernUniversität in Hagen, 2003
(postscript,
pdf)

Comparing Cartesianclosed Categories
of (Core) Compactly Generated Spaces
Martín Escardó, Jimmie Lawson and Alex Simpson
In Topology and its Applications, 143:105145, 2004.
(postscript,
pdf)
Topological domain theory involves restricting qcb spaces
to ones enjoying a form of chain completeness sufficient
for allowing domaintheoretic constructions.
The technical development of this is carried out in the
references below.

A Category of Topological Predomains
Ingo Battenfeld
Diplomarbeit. TU Darmstadt, 2004
(postscript,
pdf)

Compactly Generated Domain Theory
Ingo Battenfeld, Matthias Schröder and Alex Simpson
Mathematical Structures in Computer Science, 16:141161, 2006.
Special issue Festschrift for Klaus Keimel
M. Escardó, A. Jung and T. Streicher (eds)
(postscript,
pdf)

Two Preservation Results for Countable Products of Sequential Spaces
Matthias Schröder and Alex Simpson
Mathematical Structures in Computer Science, vol. 17, No. 1, 12pp.,
to appear 2007
Special Issue on Constructive analysis,
types and exact real numbers
H. Geuvers, M. Niqui, B. Spitters and F. Wiedijk (eds)
(postscript,
pdf)
Computational effects in topological domain theory
The various nonfunctional aspects of computation
(e.g. nondeterminism, imperative features, control facilities)
are collectively known as computational effects.
Plotkin and Power have argued that many
computational effects can be modelled using free algebras
for equational theories expressing natural equalities
between effercttriggering operations. The papers below
adapt this to topological domain theory.

Computational Effects in Topological Domain Theory
Ingo Battenfeld
Proceedings of the 22nd Annual Conference on Mathematical Foundations
of Programming Semantics (MFPS XXII)
Electronic Notes in Computer Science, 158:5980, 2006
(postscript,
pdf)

Comparing free algebras in Topological and Classical Domain Theory
Ingo Battenfeld
Submitted to journal special issue for selected papers from MFPS XXII
(postscript,
pdf)

Two Probabilistic Powerdomains in Topological Domain Theory
Ingo Battenfeld and Alex Simpson
Manuscript December 2006
(postscript,
pdf)
The last paper above also makes use of an alternative "observational"
approach to computational effects, and depends mathematically
upon the
following motivating example for the approach.

Probabilistic Observations and Valuations (Extended Abstract)
Matthias Schröder and Alex Simpson
In Proceedings of the 21st Annual Conference on
Mathematical Foundations of Programming Semantics (MFPS XXI)
Electronic Notes in Computer Science, 155: 605615 , 2006
(postscript,
pdf)
Related contributions to computability on continuous
structures
The following papers address
related issues in finding appropriate
topological structure for representing
diverse aspects
of continuous computation.

Spaces Allowing Type2 Complexity Theory revisited
Matthias Schröder
Mathematical Logic Quarterly, 50:443459, 2004
(postscript,
pdf)

Some Examples of NonMetrizable Spaces Allowing a Simple
Type2 Complexity Theory
Daren Kunkle and Matthias Schröder
Proceedings of Computability and Complexity in Analysis 2004
Electronic Notes in Theoretical Computer Science, 120:111123, 2005
(postscript,
pdf)

Computing with Sequences, Weak Topologies and the Axiom of Choice
Vasco Brattka and and Matthias Schröder
Proceedings of Computer Science Logic 2005
Springer Lecture Notes in Computer Science 3634, pp. 462476, 2005
(postscript,
pdf)

Representing Probability Measures using Probabilistic Processes
(conference version)
Matthias Schröder and Alex Simpson
In Proceedings of Second International Conference on Computability
and Complexity in Analysis
FernUniversität Hagen, Germany, Informatik Berichte, 3267, 2005
(postscript,
pdf)

Representing Probability Measures using Probabilistic Processes
(journal version)
Matthias Schröder and Alex Simpson
Journal of Complexity, 22: 768788, 2006
Special Issue on Computability, Complexity and Analysis
V. Brattka, P. Hertling, KI. Ko and H. Tsuiki (eds)
(postscript,
pdf)

Admissible Representations of Probability Measures
Matthias Schröder
Proceedings of
Computability and Complexity in Analysis 2006
Electronic Notes in Theoretical Computer Science, to appear, 2006.
(postscript,
pdf)