Axiom of Choice in General Topology

Question

I ask this question after having searched a bit the web and not having found much about the role of Axiom of Choice and General Topology.

I am trying to answer: where does the axiom of choice enters General Topology? Would it be possible to track its impact on the subject and explicit its most profound consequences? What would General Topology look without it?

Being only in my undergarduate studies I couldn't find a satisfactory answer. The fact is: I studied proofs involving AOC, in its equivalent formulation of Zorn's Lemma, both in Algebra (mainly about rings ideals and, most famously, the existence of a Hammel Basis for a vector space) and in Analysis (extension theorems of linear maps, most importantly: the Hahn Banach theorem). In a hypothetical "foundational hierarchy" of Mathematics I would naively place General Topology somehow in the middle between Logic/Algebra and Analysis (I am obviously strongly stereotyping the concept! Don't misunderstand me) and so I would expect AOC to have a grip also in this field.

Coming to what I know about General Topoloogy, it seems to me one AOC enters "only" in coverings and their properties (paracompactness, types of refinements...) which in turn found all the theorems about metrizability, metrics spaces and so on (gauges, uniform spaces, completeness). But I mean, one should expet this: we are somehow going in the direction of Analysis. There is much gneral topology beyond these topics.

I hope you can help me, maybe also through articles/books.

Answer 1

See the papers "Horrors of topology without AC, a non-normal orderable space" (van Douwen) and "continuing horrors of topology without choice" (Good and Tree)

Basic things that can go wrong in the absence of choice: $\mathbb{R}$ can be the union of countably many countable sets (so is meagre in itself and Baire's theorem fails), a sequentially continuous function on a metric space need not be continuous, etc etc. It's used in very many places, especially its countable form. We cannot define compactifications in the usual way, a lot of dimension theory becomes invalid, no nice theory of ordered spaces etc. It's certainly not confined to just covering properties; it touches almost all parts of topology and analysis.

Answer 2

Tychonoff's theorem, which states that the product of any set of compact topological spaces is compact with respect to the product topology, is equivalent to the Axiom of Choice. And Tychonoff's theorem is one of the most important theorems in Topology.

Answer 3

The proof of the Banach-Tarski paradox involves ideas from topology, geometry, measure theory, set theory and group theory. That, and a whole bunch of related results, are strongly dependent on the AOC. Stan Wagon's book offers comprehensive coverage of the subject. https://www.amazon.com/Banach-Tarski-Paradox-Encyclopedia-Mathematics-Applications/dp/0521457041/ref=sr_1_2?keywords=Banach-Tarski+paradox&qid=1558299855&s=books&sr=1-2

Len Wapner's book is a popularized discussion of the result. https://www.amazon.com/Pea-Sun-Mathematical-Paradox/dp/1568813279/ref=cm_cr_arp_d_product_sims?ie=UTF8#customerReviews