Topological Equivalents of the Axiom of Choice and of Weak Forms of Choice, by Eric Schechter

Topology Atlas Document # zaab-18.htm | Production Editor: R. Flagg

Topology Atlas Invited Contributions, vol. 1, issue 4 (1996), 60-62.

Topological Equivalents of the Axiom of Choice and of Weak Forms of Choice
by

Eric Schechter

(Department of Mathematics, Vanderbilt University, Nashville TN 37240-0001, U.S.A.)

The Axiom of Choice (AC) has many important equivalent forms in many branches of mathematics -- Zorn's Lemma, the Well Ordering Principle, the Vector Basis Theorem. In general topology, perhaps the most important equivalent is

Tychonov's Product Theorem: any product of compact topological spaces, when equipped with the product topology, is also compact.

Some other statements about product topologies are also equivalent: the product of complete uniform spaces, when equipped with the product uniformity, is also complete; the product of closures of subsets of topological spaces is equal to the closure of the product of those subsets.

The term ``constructive'' is used in different fashion by different mathematicians. For the most part, it means that we can ``find'' the object in question, and not just prove that it exists. The Axiom of Choice is the most well-known nonconstructive assertion of existence; it has important consequences for many branches of mathematics. The Axiom of Foundation (also known as the Axiom of Regularity) is also nonconstructive, but it has few applications in ``ordinary'' mathematics (i.e., outside of set theory). However, nonconstructiveness can occur not only in our axioms, but even in our reasoning: proof by contradiction is a technique of reasoning widely used in mainstream mathematics. To see that it is nonconstructive, suppose that we know that the statement "A or B" is true. From that information we cannot determine which of the two alternatives A or B is true.

The Boolean Prime Ideal Theorem and the Axiom of Dependent Choice, discussed briefly below, are known to be strictly weaker than the Axiom of Choice.

The Boolean Prime Ideal Theorem (PI) also has many important equivalents in many branches of mathematics -- the Stone Representation Theorem, the Compactness and Completeness Principles of Logic. Some of its equivalents in topology are:

Any filter is contained in an ultrafilter.

Any net has a universal subnet.

A topological space is compact if and only if every ultrafilter (equivalently, every universal net) in it is convergent.

Any completely regular Hausdorff space has a Stone-Cech compactification.

Any product of compact Hausdorff spaces is compact.

For any set X, the set 2^X (with the product topology) is compact.

The Axiom of Dependent Choice (DC) has a few interesting equivalents. Surely the most important of these is the Baire Category Theorem.

Although the term ``constructive'' is used in different fashion by different mathematicians, the Axiom of Dependent Choice is the strongest form of choice that is widely held to be constructive. Thus, we obtain a first approximation to constructive mathematics if we replace ZF+AC (conventional set theory) with ZF+DC. However, this is only a first approximation, since it still permits the Axiom of Foundation and proofs by contradiction. In the minds of some mathematicians (e.g., the present author), there is a closeness between the notion of ``explicitly constructible example'' and the notion of ``object whose existence can be proved using just ZF+DC''. The latter notion has the advantage that it is very precise and can be studied with some success using fairly simple mathematical tools, discussed below.

There is a strong analogy between measure theory and general topology, which is developed in J. C. Oxtoby's book (Measure and Category, 2nd edition, 1980, Springer) and more recently in J. C. Morgan's book (Point Set Theory, 1990, Marcel Dekker). The Lebesgue-measurable sets correspond to the sets with the Baire property -- i.e., the sets which are the symmetric difference of an open set and a meager set. A couple of very weak consequences of the Axiom of Choice are the existence of (i) subsets of R which are not Lebesgue measurable, and (ii) subsets of R which lack the Baire property. The existence of those two kinds of pathological objects can be proved using other assumptions instead of the Axiom of Choice; in particular, the existence of those pathological objects follows from the existence of many other pathological objects which are classical in the literature.

Let BP be the statement that ``every subset of R has the Baire property''; this is a very strong negation of the Axiom of Choice. In 1984 Shelah proved that the consistency of ZF implies the consistency of ZF+DC+BP. Therefore a subset of R which lacks the Baire property is not ``explicitly constructible'', in the sense mentioned above -- i.e., it cannot be proved to exist using just ZF+DC. Therefore many other classical pathological objects are not ``explicitly constructible''. Thus, Shelah's consistency result gives us an easy way to prove that many classical pathological objects lack examples.

All of the topics sketched above are studied in greater detail in my book, Choice, Completeness, Compactness. I have posted a web page which advertises the book. The web page includes some additional information which may be of interest: lists of equivalents of AC, PI, DC, and HB (the Hahn-Banach Theorem, another weak form of choice), and a chart showing the relations between some of the weak forms of choice.

Received by the editors: January 16, 1996; Revised: January 17, 1996.