Lectures on Boolean Algebras by Paul R. Halmos

Author:Paul R. Halmos
Language: eng
Format: epub
Publisher: Dover Publications, Inc.
Published: 2018-10-14T16:00:00+00:00

§17. Boolean spaces

We know by now that not every Boolean algebra is isomorphic to the field of all subsets of some set. In the next section we shall prove that every Boolean algebra is isomorphic to some field of subsets of some set. In order to get a usable description of what kind of fields and what kind of sets are needed, we proceed now to introduce a rather special category of topological spaces.

A Boolean space is a totally disconnected compact Hausdorff space. There are several possible definitions of total disconnectedness, but, as it turns out, they are all equivalent for compact Hausdorff spaces. The most convenient definition for our algebraic purposes is the one that demands that the clopen sets constitute a base. Explicitly: a Boolean space is a compact Hausdorff space with the property that every open set is the union of those simultaneously closed and open sets that it happens to include.

For Boolean spaces, as for every topological space, it is true that the class of all clopen sets is a field. The field of all clopen sets in a Boolean space X is called the dual algebra of X.

The simplest Boolean spaces are the finite discrete spaces. Since every subset of such a space is clopen, the dual algebra of each finite Boolean space is a finite Boolean algebra. Since every finite Boolean algebra is isomorphic to the field of all subsets of some (necessarily finite) set (see §16), it follows that every finite Boolean algebra is isomorphic to the dual algebra of some discrete Boolean space.

A less trivial collection of examples consists of the one-point compactifications of infinite discrete spaces. Explicitly, suppose that a set X with a distinguished point x0 is topologized as follows: every subset of the complement of {x0} is open, but a set containing x0 is open if and only if its complement is finite. It is easy to verify that the space X so defined is Boolean; a subset of X is clopen if and only if it is either a finite subset of X – {x0} or a cofinite subset (of X) containing x0. The dual algebra of X is isomorphic to the finite-cofinite algebra of X – {x0}.

The set 2 is a Boolean algebra; from now on it will be convenient to construe it as a topological space as well, endowed with the discrete topology. For an arbitrary set I, the set 2I of all functions from I into 2 (equivalently: the Cartesian product of copies of 2, one for each element of I) is a topological space (product topology); it is well known that that space is compact and Hausdorff (Tychonoff’s theorem). We shall denote the value of a function x in 2I at an element i of I by xi. The sets of the form {x 2I : xi = δ}, where i I and δ 2, constitute a subbase for 2I; finite intersections of them constitute a base. Since the complement of each set of

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