Abstract
This chapter discusses basic concepts of Čech–Stone remainders of discrete spaces. The study of Čech–Stone remainders has long been a major theme in set-theoretic topology. The chapter begins with discussing two problem: (1) “Is it consistent that w* is homeomorphic to w*1?” and (2) “Is it consistent that the Boolean algebras P(ω)/fin and P(ω1)/[ ω1]<ω are isomorphic?” Here w*1 refers to the Čech–Stone remainder of a discrete space of cardinality ω1. In the absence of the Axiom of Choice (AC) the two problems are not equivalent: what passes for the Čech–Stone remainders could be empty, while the quotient algebras are both uncountable. It would be interesting if the Boolean algebra version had a positive answer in ZF, while the answer to both versions is negative in ZFC. Some basic facts about the Čech–Stone compactifications of discrete spaces are also discussed in the chapter. Some consequences and problems related to axiom Ω are also discussed.
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