Stone space

In topology, and related areas of mathematics, a Stone space is a non-empty compact totally disconnected Hausdorff space.[1] Such spaces are also called profinite spaces.[2] They are named after Marshall Harvey Stone.

A form of Stone's representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to the Boolean algebra of clopen sets of a Stone space. This isomorphism forms a category-theoretic duality between the categories of Boolean algebras and Stone spaces.

Equivalently[3], Stone space is a topological space such that:

  • Compact, totally separated;
  • Compact, , zero-dimensional;
  • Coherent and Hausdorff.

References

  1. ↑ Hazewinkel, Michiel, ed. (2001) [1994], "Stone space", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
  2. ↑ Stone space in nLab
  3. ↑ "Boolean Algebra". orion.math.iastate.edu.
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