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Boolean Algebras

In: Extensions and Absolutes of Hausdorff Spaces

Author

Listed:
  • Jack R. Porter

    (The University of Kansas, Department of Mathematics)

  • R. Grant Woods

    (University of Manitoba, Department of Mathematics)

Abstract

In this chapter, a special type of lattice, called a Boolean algebra, is investigated. The set of clopen subsets of an arbitrary space is shown to be a Boolean algebra with respect to unions and intersections, and any Boolean algebra is shown to be isomorphic to the set of clopen sets of a unique compact, zero-dimensional space. This correspondence establishes a duality between the class of Boolean algebras and the class of compact, zero-dimensional spaces. This duality result is used to show that certain (atomless) countable Boolean algebras are isomorphic to each other and to the set of clopen sets of the Cantor space. In 3.4 we focus our attention on complete Boolean algebras (Boolean algebras in which arbitrary suprema and infima exist) and show that every Boolean algebra is isomorphic to a subalgebra of a complete Boolean algebra. We close the chapter by discussing Martin’s Axiom, and some topological and combinatorial applications of it, in 3.5.

Suggested Citation

  • Jack R. Porter & R. Grant Woods, 1988. "Boolean Algebras," Springer Books, in: Extensions and Absolutes of Hausdorff Spaces, chapter 0, pages 155-237, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4612-3712-9_3
    DOI: 10.1007/978-1-4612-3712-9_3
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