To bw, or not to bw: that is the question.
by
Masaru Kada
RISE, Waseda University
Coauthors: Kazuo Tomoyasu and Yasuo Yoshinobu

It is known that the Stone-Cech compactification bX of a metrizable space X is approximated by the collection of Smirnov compactifications of X for all compatible metrics on X [Woods, 1995]. Furthermore, if X is locally compact and separable, the statement holds also for Higson compactifications [Kawamura-Tomoyasu, 2001]. Now we let X=w and ask the following two questions.

(1) How many "smaller" Smirnov or Higson compactifications do we actually need to approximate bw?

(2) Can we approximate bw by an "increasing chain" which consists of Smirnov or Higson compactifications?

To state these questions precisely, we define the following cardinals.

Let sp be the smallest size of a set of compatible metrics on w such that, the set of corresponding Smirnov compactifications approximate bw but any finite subset does not suffice, and hp the corresponding cardinal for Higson compactifications.

Let st be the shortest length of a sequence of compatible metrics on w such that, the corresponding Smirnov compactifications form an increasing chain (with respect to a natural ordering among compactifications), each one is smaller than bw, and the whole chain approximates bw. If no such sequence exists, we write st=¥. ht is defined analogously for Higson compactifications.

We will present ZFC results and consistency results on the relationship among these cardinals and other known cardinal characteristics of the reals.

Paper reference: arXiv:math.GN/0412547

Date received: March 9, 2005