Topology et al. Seminar: Wednesday, 11 November 2009

Cardinal Invariants for kappa-Box Products

Speaker: Wistar Comfort, Wesleyan University
Abstract: This derives from joint work with Ivan Gotchev. The symbols w, d and S denote density character, weight and Souslin number, respectively, this last defined as follows: for a space X, S(X) is the least cardinal alpha such that X admits no family of alpha-many pairwise disjoint nonempty open subsets. [Remark: Always d(X) <= w(X), S(X) <= (d(X))^+.] Now, given a set {X_i : i in I} of nontrivial spaces and denoting by X_I their usual topological product, consider these basic results from General Topology. 1. w(X_I) = max{|I|, sup{w(X_i) : i in I}}. 2. [Hewitt-Marczewski-Pondiczery] If alpha >= omega, |I| <= 2^alpha and each d(X_i) <= alpha, then d(X_I) <= alpha. 3. If \alpha >= omega and each d(X_i) <= alpha, then S(X_I) <= alpha^+. 4. Let \alpha := sup{S(X_F : F subseteq I, F is finite}. Then S(X_I) = alpha if alpha is regular, S(X_I) = \alpha^+ otherwise. The authors generalize those and other familiar cardinality results about product spaces X_I to spaces of the form (X_I)_kappa, which is X_I with the kappa-box topology (basic open sets are restricted in (alpha^+)^2_alpha play a prominent role. (Motivational combinatorial test question: Given a sequence of finite sets, is there a subsequence whose pairwise intersections coincide?)

Time and place: 04:15 PM, ESC 638.