GeneralTopology


Topology

Topology is often described as the field that lets you characterize the difference between a saucer and a coffee cup, and identify the cup with a doughnut. The coffee cup and doughnut have "holes" and the saucer does not.

A little more seriously topology studies continous change ( which is not the same thing as continual change, although some confuse the two). It shows how things can change smoothly while remaining the same.

General Topology

Some fields of topology (such as algebraic topology and differential topology) specialize in topology involving very special added structure, really intended to study fields other than topology. General topology sticks to things that are rather directly derived from the basic axioms of topology.

My own particular work, some of which is listed below (mostly done long ago) was to find very general structure where none had been found before.

My later work discovered some specific simple topological spaces which could act as "prime numbers" for constructing and characterizing very general classes of spaces. That is, any space in the class can be constructed from my spaces, which is the easy part, but if my spaces are constructed from other spaces one of the other spaces must be one of mine: that is the hard part, the primeness, and is something very unusual in topology.

By the time I made those discoveries my teaching interests and resonsibilities had changed dramatically into distributed computing, and I had begun to chair (for 21 years) a department which had begun to focus on computing, as well as mathematics and statistics. Most of my time was spent on those matters, but I named the main computer on which I work spectral to remind myself that someday I should get back to the work, finish it off, and publish it. That finishing is now completed, and the publishing is ready to be done.

Spectral Spaces of Rings

There is a very important type of object called a "commutative ring with unit", that has been and still is very much an object of study by many mathematicians in many areas. From such a ring one can derive what is often called its "strucure space" or its "space of prime ideals" or as I will call it its "spectral space". In 1969 a characterization was given by Melvin Hochster of the topological spaces that could be spectral spaces of rings. He did this by giving a highly theoretical and complex construction of a ring to go along with such a space.

A few years after that as I discovered the "irreducible" spaces mentioned above, I began to realize that they would have to be the factors from which spectral spaces are uniquely built, and therefore that their rings would have to be very fundamental as well. But the precise construction of such rings eluded me.

A little before 2000 I discovered how to construct those rings in a special case, and already knew from topology that they were the only ones really needed to rather simply construct the other rings. So I focussed on them and began to really understand what was involved. There is now a very straightforward way to do all of this and that is what my current work, more or less sandwiched between my other duties, has done.

Outline of my completed work on the topic

A Few Principal References for Spectral Spaces