Research Overview

Please note that many links to articles and preprints are broken. My apologies.

During the 1990's, I worked on creating and developing higher-level language for set-theoretic topology. Many proofs in topology are technically difficult and yet are examples of standard technique. The purpose of higher-level language is to make such difficulty disappear.
Proofs used in the construction of topological spaces can be difficult yet standard. The higher-level notion of resolution greatly simplifies these proofs and strengthens our understanding. See Theory of Construction . The higher-level use of partial orders has a similar effect on the construction of non-Tychonoff spaces. See Between Partial Orders and Topological Constructions.

Another such type of proof is the common ``closing-off'' argument. Elementary submodels shorten such arguments and clarify the essential ideas. The related higher-level notions of finite hull and scheduled relativization greatly simplify many other difficult arguments.
See Logic for an overview and see A Guide to Finite Hulls for background material and work in progress.

I have also been examining the applications of topological methods to the mathematical sciences. Rather than wait for practitioners in other fields to bring their topological problems to the specialists, I have searched for these problems in their literature. These applications do not occur as isolated curiosities but rather cohere in a deeply interconnected maze. These applications are in Finite Combinatorics , especially the study of partial orders, Algebra , especially in category theory, Analysis , in hyperspaces, function spaces, and measure spaces as well in the study of compacta in Banach spaces and boundaries in Euclidean space , and in more distant areas such as Metric Geometry and Computer Science . Meanwhile I have continued to study traditional set-theoretic topology.

This work has been presented in invited lectures at many conferences.
At present, only a small selection of reprints/preprints are available online. This will gradually be improved. The list of publications will eventually be expanded to include all unpublished papers .
Here is a list of questions which will be gradually enlarged from my personal files.