Daniel was my first masters student, completing his M.S. in June 2010. I co-advised him together with my friend and colleague Ramiro de la Vega, at the Universidad de los Andes, in Bogota. The following picture is from his Facebook profile.
Daniel’s thesis, Metric spaces on under the axiom of determinacy (in Spanish), is part of a vastly unexplored field: general topology in the absence of choice.
Work on this area has been mostly about highlighting pathologies, illustrating how vastly different results can be when we keep standard definitions but work in setting where the axiom of choice fails badly. Even in the setting of the real numbers with the standard topology, things may not work as expected: Every set of reals may be Borel (in fact ), there may be Borel infinite Dedekind-finite sets, etc. In his PhD thesis at UC Berkeley, Apollo Hogan showed that, instead, we can carry out a systematic and detailed study of general topology if instead of dealing with arbitrary models with absurd failures of choice, we concentrate on settings where the absence of choice is compensated by a rich combinatorial structure. Specifically, Apollo (who was a student at Berkeley at the same time I was there) considered topology under the axiom of determinacy. Daniel’s thesis is a survey of some of the results discovered by Apollo.
Daniel begins by reviewing some of the basic consequences of , the axiom of determinacy. To state , we need to consider certain ideal games between two players, that I will just call and . In all these games the format is the same: players and alternate with playing first. In each turn, the corresponding player picks a natural number, repetitions being allowed, and both players having knowledge of all the moves both have made previously. They play for infinitely many rounds. At the end, a sequence of natural numbers has been produced, with being the numbers picked by player , and being the ones picked by . We have one of these games for each set , where is the set of all infinite sequences of natural numbers. In the game associated to such an , player wins iff the sequence is in . Otherwise, player is the winner.
A strategy for player is a function that tells player what to play each time. Formally, this is just a function from the set of finite sequences of numbers to . A strategy for is defined similarly. We say that a strategy for is winning if and only if player wins the game whenever they play following the strategy. That is, in any such game we have , etc, and at the end we have that . Winning strategies for player are defined similarly. We say that is determined iff one of the players has a winning strategy.
It is easy to give examples of determined sets . Using the axiom of choice, we can give examples of undetermined sets, but deep theorems in descriptive set theory indicate that no undetermined set can be particularly simple. For instance, it is a celebrated theorem of Martin that all Borel sets are determined. Here, is made into a topological space by taking the product topology of countably many copies of the discrete set .
The axiom of determinacy is the statement that all are determined. In particular, this statement contradicts the axiom of choice. See here for slides of a talk I gave a few years ago containing a quick introduction to the subject.
The short remainder of this post (after the fold) is by necessity more technical.
Since contradicts choice, it is studied in the context of . It is not completely incompatible with choice, of course. For instance, it implies countable choice for sets of reals. If determinacy holds, it holds in , the constructive closure (in the sense of Gödel) of the set of reals. This is the smallest transitive model of that contains all the reals and all the ordinals. The relevance of this remark is that the consequences of are much better understood when studied under the assumption that . For example, Kechris proved that in this case we have not just countable choice for sets of reals, but in fact dependent choice (this is strictly stronger than countable choice for all sets, which in turn is strictly stronger than countable choice for sets of reals). This means that we can carry out a decent theory of analysis under the assumption of determinacy, and we would see no changes at the classical level. On the other hand, the presence of determinacy actually eliminates some of the (perhaps pathological) consequences of choice such as the existence of sets of reals that are not Lebesgue measurable.
One of the remarkable properties of determinacy is that it implies the existence of large cardinals. For example, Solovay proved that and are measurable. This gives us in particular that exists for all reals . Daniel reviews the theory of sharps from a (very) classical point of view (as EM blueprints). He also verifies that, under determinacy, every subset of is constructible from a real.
With these preliminaries out of the way, Daniel proceeds to examine the theory of metric spaces on ordinals under determinacy. Following Hogan, he shows that any metric space on embeds into the product space , where carries the discrete topology. Using this, he shows that any such space is the union of countably many discrete closed sets.