Cellularity of Growths

Abstract

In this chapter we consider relationships between the existence of families of open subsets of a space X and the cellularity of X* =βX\X. Recall that the cellularity of a space Y is the smallest cardinal number m for which each pairwise disjoint family of non-empty open sets of Y has m or fewer members. The density of a space Y is the smallest cardinal number which can be the cardinal number of some dense subspace of Y. It is clear that the density of a space is at least as great as the cellularity. Most of the results will take the form of providing a lower bound for the cellularity of X* by demonstrating the existence of families of pairwise disjoint open sets in X*. The methods used will be reminiscent of that used in Chapter 3 to show that the cellularity of ℕ* is c. In the last sections of the chapter, we will show that there exists a point in ℕ* which belongs to the closure of each member of a family of c pairwise disjoint open sets of ℕ*.