Math 738N (MWF 12:20 - 1:10, in LeConte 303B) Set-Theoretic Topology

Spring 2012

Professor: Peter J. Nyikos

Office: LeConte 406

Phone: 7-5134

Email: nyikos @ math.sc.edu

Special office hours for Tuesday, April 24: 2:00- 4:30; Wednesday, April 25: 1:20- 1:50; Thursday, April 26: 2:45 - 3:15 and 4:30 - 5:45; Friday, April 27 and Monday, April 30: 2:15 - 4:45; Tuesday, May 1: 8:30 - 11:40 and 2:00 - 2:30; Office hours for rest of exam period tba.

Syllabus: Chapters I - VI of Rudin's booklet Lectures on Set Theoretic Topology supplemented by notes from a textbook in progress, in conjunction with photocopies of pages from topology textbooks to make the booklet self-contained, and from Handbook of Set-Theoretic Topology to bring it sufficiently up to date.

Learning outcomes: A student who successfully completes MATH 738N will be able to understand and apply partition calculus theorems to the study of cardinal invariants of topological spaces, and to use the simpler axioms derived from the Axiom of Constructibility to the construction of Souslin trees and unusual topological spaces on the one hand, and to prove theorems using versions of Martin's Axiom which imply that such trees and spaces do not exist, but also imply that certain other unusual spaces do exist,

This is a course in the applications of modern set theory to some of the most basic concepts in topology. Modern set theory has enabled us to solve some long-outstanding problems about these concepts, and to show that some others cannot be settled by the usual axioms (in other words, all generally accepted axioms that underlie all mathematics). The course shows the usual axioms in action, along with special axioms that can be neither proved nor disproved by the usual axioms.

The text for this course is a booklet by Mary Ellen Rudin, Lectures on Set Theoretic Topology. Although it was published in 1977, and so omits a great deal of exciting research that has taken place since then [see links below for a few examples], it is still the only work in print that is suitable as a textbook in the subject at this beginning level. I am currently working on a textbook that would be up to date, and handouts are being be provided frequently from the parts that are in a sufficiently polished form.

Special axioms which play a prominent role in this course include the Continuum Hypothesis, Martin's Axiom, the Product Measure Extension Axiom, and G\"odel's Axiom of Constructibility. No prior familiarity with these axioms is assumed. About half of Rudin's booklet will be covered during the course, and the rest is the foundation for a follow-up course, which is being offered for Fall 2012.

Set theory has been completely revolutionized since 1963, when Cohen completed the work that Godel began in the 1930's, in solving Hilbert's First Problem. This asked whether every set could be well-ordered, and whether every subset of the real line is either countable or could be put into one-to-one correspondence with the entire real line. The immediate outcome was that the generally accepted axioms of set theory (ZF) are inadequate to settle either of these statements. But far more importantly, the methods used by Godel and Cohen have led to a remarkable assortment of new techniques for tackling some of the most elementary questions in various areas of mathematics.

Set theoretic topology is perhaps the best introduction to these techniques besides set theory itself. Some hint of the power and usefulness of these techniques may be found in my statement of research interests and in an electronic article in Topology Atlas.