Character

Abstract

First note that we can define χA as a sup; namely, for any ultrafilter F on A let χF = min{|X| : X is a set of generators of F}—then χA = sup[χF : F is an ultrafilter on A]. Clearly then, by topological duality, χ(A×B) = sup(χA, χB). For a weak product we have $\chi \left( {{{\prod }_{{i \in I}}}{{A}_{i}}} \right) = \max \left( {{{2}^{{\left| I \right|}}},{{{\sup }}_{{i \in I\chi }}}{{A}_{i}}} \right)?$. To show this, it suffices to show that χF = |I| for the “new” ultrafilter F. This ultrafilter is defined as follows. For each subset M of I, let x _M be the element of Π _i∈I A _i such that x _M i = 1 if i ∈ M and x _M i = 0 for i ∉ M. Then F is the set of all $y \in \prod _{{i \in I}}^{w}{{A}_{i}}$ such that x _M ≤ y for some cofinite subset M of I. So, it is clear that χF ≤ |I|. If X is a set of generators for F with |X| < |I|, then there is a y ∈ X such that y ⊆ x _M for infinitely many cofinite subsets M of I; this is clearly impossible.