The Fourth Head of βℕ

Abstract

This chapter explains some concept related to βℕ. ℕ is the space of natural numbers with the discrete topology, and βℕ is Čech-Stone compactification. In his introduction to βℕ, Jan van Mill called it a three-headed monster. The first head shows under the Continuum Hypothesis (CH) and that it is smiling and friendly because CH easily resolves problems about βℕ, more precisely, as easily as solutions to problems about βℕ get. The second head is the "ugly head of independence" as Paul Erdos used to call it. The smallest, third, head is the ZFC-head of βℕ. It provides those few facts about βℕ that can be resolved without applying additional set-theoretic axioms. Ever since Shelah's groundbreaking results came, there is emergence of the fourth head of βℕ based on a coherent theory of βℕ deduced from forcing axioms (or Ramseyan axioms) with strong rigidity phenomena for βℕ and similar Čech-Stone compactifications. The basics of trivial continuous maps also reviewed in the chapter. Concepts of rigidity phenomena for quotients P(ℕ)/. I are also presented in the chapter.