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.