D-17 - The Čech-Stone Compactification

Abstract

Stone applied his theory of representations of Boolean algebras to various topological problems. The Čech construction proceeds by taking the Boolean algebra B generated by all cozero sets and all nowhere dense subsets of X. The compactification constructed by Čech and Stone is now called the Čech–Stone compactification. A subspace A is C-embedded (C*) in a space X if every (bounded) real-valued continuous function on A can be extended to a continuous real-valued function on X. Spaces with an easily identifiable Čech–Stone compactification are pseudocompact. A perfect map is continuous, closed, and with compact fibers. Properties that are carried over Čech–Stone compactification are usually of a global nature. —for example, connectedness, external disconnectedness, basic disconnectedness, and the values of the large inductive dimension (for normal spaces) and covering dimension. According to the Continuum Hypothesis, all separable spaces have remote points; however, in the side-by-side Sacks model, there is a separable space without remote points. Many spaces with the countable chain condition have remote points and it is unknown whether there is such a space without remote points.