© 1996, Topology Atlas

Of the axioms of set theory, the Axiom of Choice is certainly the one that has excited the most interest amongst non-set-theorists. Although its assumption leads to some apparently counterintuitive conclusions, such as the well-ordering of the reals and the Banach-Tarski paradox, many important and widely accepted results are known to require at least some fragment of choice: Tychonoff's theorem and the existence of maximal Abelian subgroups, both of which are known to be equivalent to the Axiom of Choice; the Hahn-Banach theorem, which follows from the Boolean Prime Ideal theorem (BPI) and implies both the Banach-Tarski paradox and the existence of sets with no Lebesgue measure; the compactness theorem of first order logic, which is equivalent to the BPI; the Baire category theorem, which is equivalent to the Axiom of Dependent Choices.

In general topology the use of choice passes almost without notice and, whilst counterintuition is no reason to suppose falsity (quantum theory, for example, is certainly not intuitive), it is obviously important to know when and how choice is used. This is the main aim in studying 'topology without choice.'

There seem to me to be three broad and connected areas for study:

Effective versus non-effective topology.

Is a toplogical result effective or does it require some form of choice? The most famous example of a non-effective topological theorem is Tychonoff's theorem which was shown by Kelly to be equivalent to the Axiom of Choice. Other examples of non-effective topology include Urysohn's Lemma (Lauchli), the normality of LOTS (van Douwen) and Stone's theorem that metric spaces are paracommpact. Using permutation groups, particularly nice models of set theory can be constructed in which it is very clear how choice fails in these examples. Examples of effective topological results include the Heine-Borel theorem and Urysohn's metrization theorem.

Effectivization of topology.

Can one state a non-effective result in a stronger form that is effective? The Hahn-Banach theorem holds effectively for separable Banach spaces. Comfort, Johnstone and Loeb all give effective versions of Tychonoff's theorem by strengthening the notion of compactness or the nature of the family of compact sets. One can do the same for Urysohn's Lemma. Roughly speaking, one is trying to change the positions of universal and existential quantifiers.

Topological structure and choice.

How much choice does a particular result require and how much does it imply? Brunner has shown, for example, that if every Hausdorff space has a well-orderable dense subset or if any cover of any Hausdorff space has a pointed refinement, then the Axiom of Choice holds. Herlich has considered simliar questions for Tychonoff's theorem.