Seminar in Topology and Set Theory, Fall 2007

Unless otherwise stated, seminars are Wednesdays 1:30--2:30 in LeConte 312

September 12

Speaker: Peter Nyikos

Title: The Stone-Cech compactification of the natural numbers N

Abstract: One way to define the Stone-Cech compactification betaN of N is to say that it is the unique compact Hausdorff space X in which N is a dense subspace and which has the property that every bounded real-valued function from N extends to a continuous function on all of X. In this formulation it is easy to show that the Banach space l_infinity can be identified with the space of continuous functions on betaN with the sup norm.

An "internal" characterization of betaN is explained in this talk. It characterizes betaN as the space whose points are the ultrafilters on N and whose topology is given in a certain simple way using the subsets of N. The connection between the two characterizations will be given in the detail that time permits. No prior knowledge of ultrafilters is assumed in this first seminar talk.

September 19

Speaker: Peter Nyikos

Title: The Stone-Cech compactification of the natural numbers N, Part II

Abstract: A continuation of the first talk, with a handout provided which includes everything needed from the first seminar.
Main Theorem: betaN is compact, but not sequentially compact.

No seminar on September 26

October 3

Speaker: Peter Nyikos

Title: The Stone-Cech compactification of the natural numbers N, Part III

Abstract: This lecture takes a different tack from the first two, treating betaN (the Stone-Cech compactification of N) axiomatically. It defines betaN as a Hausdorff compactification of N in which any two disjoint subsets of N have disjoint closures. Using this characterization, it is very easy to show that betaN is not sequentially compact, and not hard to show that every convergent sequence in betaN is eventually constant. The following unsolved problem is discussed: is there an infinite compact space on which every convergent sequence is eventually constant, yet does not contain a copy of betaN?

October 10

Speaker: Peter Nyikos

Title: The Baire Category Theorem applied to normality

Abstract: The Baire Category Theorem has applications that go well beyond the compact metric spaces that it is literally about. One is the proof that certain modifications of metric spaces are not even normal. (A normal topological space is one in which disjoint closed sets can be put into disjoint open sets.) One example is the Sorgenfrey plane, another is the Moore plane. In either case we can exhibit a pair of disjoint closed sets that violate the normality criterion, using the Baire category theorem to confirm the violation.

Another example, known as Heath's tangent V space, is metacompact (but also not normal, by a very similar argument to the other two). The significance of this example is explained. No attendance at previous seminars is assumed.

October 24

Speaker: Peter Nyikos

Title: Normal Moore spaces

Abstract: Every metrizable space is a normal Moore space, and the question of whether the converse is true was one of the best-known unsolved problems in general topology for the better part of a century. It has a very unusual status: counterexamples exist under various set-theoretic assumptions including the Continuum Hypothesis and Martin's Axiom, but if one assumes the consistency of some very large cardinal numbers, then every normal Moore space is metrizable--and it is known that very large cardinals are required! This status will be explained and two of the Martin's Axiom examples described; they are normal subspaces of spaces which were shown to be non-normal two seminars ago. However, no prior attendance at the seminar is assumed.

One key to the large cardinal results is an axiom which implies, among other things, that Lebesgue measure on the real line can be extended to a countably additive (but, of course, not translation-invariant!) measure on all the subsets of the real line.

November 14

Speaker: Matthew Gamel

Title: On the Metrizability of the Weak Topology on a Banach Space

Abstract: I will essentially furnish a proof of the following theorem: the weak topology on a Banach space X is metrizable if and only if X is finite dimensional. I will try to make this talk self contained insofar as prerequisite knowledge in functional analysis will be minimal. Some cornerstone results in the subject will be needed but will be explicitly stated.