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?)
