Speaker: Peter Nyikos

Title: Measurability and ultrafilters

Measures are generally taken to be countably additive. In the first seminar this was modified to allow for finitelyadditive measures. This second seminar explores the relationship between countably additive measures and ultrafilters. There is a natural one-to-one correspondence between ultrafilters with the countable intersection property on a set X and 2-valued, countably additive measures defined on all subsets of X. If the measure is nonatomic (i.e. every singleton has measure 0) then X has to be enormous. Some idea will be given in the seminar of just how enormous such an X has to be.

There will be a handout covering all that you need to know from the first seminar, and it will be reviewed for the benefit of anyone who missed it. In particular, no prior knowledge of ultrafilters is assumed.

Speaker: Peter Nyikos

Title: 2-valued Measures and Measurable Cardinal Numbers

Abstract: A 2-valued measure on a set X is a countably additive measure u such that u(X) = 1 and u(A) is either 0 or 1 for all subsets A of X. An issue with profound implications for set theory is what kinds of 2-valued measures can be defined for all subsets of a set X. There are the trivial ones, in which some singleton {p} has measure 1 and a subset of X has measure 1 iff it includes p.

It is consistent with the usual (ZFC) axioms of set theory that they are all trivial. Any set on which there are others has to be enormous -- so enormous that there is some question as to whether it is inconsistent to assume that they exist at all. In this seminar some small hint will be given as to just how enormous even the smallest of them has to be. In other words, we give some idea as to how large the first "measurable cardinal number" has to be.

Speaker: Peter Nyikos

Title: Large Cardinal Numbers and Haar measure

Abstract: The smallest measurable cardinal (if such a thing exists!) is the smallest cardinal number for which there is a 2-valued, countably additive measure u on a set X, defined on all subsets of X, such that u(X) = 1 and u(A)= 0 for all finite (hence all countable) subsets of X. It will be shown that if this cardinal number is greater than |X|, then so is 2^|X|. Thus, in particular, the real line carries no such measure.

However, if it is consistent that there is a measurable cardinal number, then it is also consistent that Lebesgue measure on R can be extended to a countably additive [though not translation-invariant!] measure defined on all subsets of R. A generalization of this to Haar measure, using strongly compact cardinals, will be explained, along with the concepts of strongly compact cardinals and Haar measure themselves.

Speaker: Peter Nyikos

Title: Introduction to topological games

Abstract: A topological game is a game played on a topological space with infinitely many moves made by two players who have opposing objectives. The first game we will consider is played using open intervals of the real line, so no prior knowledge of topology is assumed. It is called the Banach-Mazur game. A subset A of the real line is given, and players take turns playing open intervals of the real line, each one inside the last one played by the opponent. One player wants the intersection of the intervals to contain at least one point of A; the other does not.

The concept of a tactic, and of various kinds of strategies will be explained. Which (if any) player has a winning strategy in this game, depends on the nature of the set A.

Speaker: Peter Nyikos

Title: A topological game involving convergence Abstract: A topological game is a game played on a topological space with infinitely many moves made by two players who have opposing objectives. This seminar features a topological game introduced by G. Gruenhage. The two players, named "the hero" and "the villain" for convenience, play a game in which a point p ("the jail") of a topological space X is fixed throughout the game. The hero picks neighborhoods of p on each turn, and the villain picks a point from the neighborhood the hero picked on his last turn. The hero wins if the sequence of points chosen by the villain converges to the jail. The question of which (if either) player has a winning strategy in this game tells a lot about the topology of X. There is a handout explaining enough for those who missed last week's talk.

Speaker: Peter Nyikos

Title: Topological games and trees

Abstract: A topological game is a game played on a topological space with infinitely many moves made by two players who have opposing objectives. This seminar features a pair of closely related topological games and some trees that are very relevant to them. One is the "decision tree" for a given strategy of either player, and two are specific trees that are relevant as to which player has a winning strategy in either of the games.

In both games, two players, named "the hero" and "the villain" in the first game and "Player I" and "Player II" in the second game, have opposite intentions about a point p of a topological space X. The hero of one game and Player II in the other game wish to have a sequence of points x_n converge to p, while the opposing players want the sequence not to converge to p.

It is the villain who picks the points in the first game, and Player II who picks them in the second game, but the same points p in the same spaces give the hero and Player II a winning strategy, while the spaces and points that give Player I and the villain a winning strategy are also the same.

The question of which (if either) player has a winning strategy in either game tells a lot about the topology of X.

There will be a handout explaining enough for those who missed the previous talks in this series of seminars.

Speaker: Peter Nyikos

Title: The equivalence of two topological games

Abstract: A topological game is a game played on a topological space with infinitely many moves made by two players who have opposing objectives. This seminar features a pair of closely related topological games that were introduced last week. [See description for last week, above.]

The one-point compactification of an Aronszajn tree [to be explained in the seminar] is one space where neither player has a winning strategy.

There will be an updated handout explaining enough for those who missed the previous talks in this series of seminars.

Speaker: Peter Nyikos

Title: Normality in products and a topological game

Abstract: A topological game provides the answer to a question posed by the famous Japanese mathematician Kiiti Morita: which normal spaces have a normal product with all metric spaces?

The game consists of one player choosing an closed set F_n on the nth move, while the other chooses an open set U_n containing F_n. Both sequences have to be descending, i.e. F_{n+1} is a subset of F_n, etc.

The first player wins if the intersection of the F_n is empty but the intersection of the U_n is not empty. Otherwise the second player wins. A Morita P-space is a space on which the second player has a winning strategy, and the normal Morita P-spaces are the answer to the question that Morita answered.

A simple modification of the real line gives a normal space that is not a Morita P-space. Several ways of showing this will be outlined.

This is in contrast to the game-free topic of which descending sequences of closed sets with empty intersection can be followed down to the empty set by open sets. This characterizes normal spaces whose product with the closed unit interval is normal, and normal spaces that do not meet this requirement are very complicated.

There will be a handout explaining enough for those who missed the previous talks in this series of seminars.