A-4 - Cardinal Functions, Part II

Abstract

This chapter introduces several topics on cardinal functions. Some of them are rather set theoretical, where set theoretical notions such as Continuum Hypothesis (CH), MA (Martin's Axiom), large cardinals, the forcing method, V equal to L and others, play important roles. The cardinal function π-weight is interesting not only from the topological point of view, but also from the Boolean algebraic point of view. A family ▪ of non-empty open sets of a space X is called a π-base if for any non-empty open set V of X, there is U ∈▪ with V ⊂ U. In terms of the theory of Boolean algebras, π-weight is “the density” (in the sense of Boolean algebras) of the Boolean algebra RO(X) of all regular open sets of X. In this manner, results on topologies can be translated to results on Boolean algebras and vice versa. A family ▪ of non-empty open sets of X is called a local π-base of x in X if for every neigh borhood V of x; there is U ∈▪ with U ⊂ V.