18
$\begingroup$

Topological games and some similar infinite games seem to be often used used as a tool in some areas of general topology, but also some other areas, such as Ramsey theory, filters, etc. Probably the best known are Banach-Mazur and Choquet game, but many other games of similar nature have been studied. Various questions about topological games and their winning strategies have also been asked on this site. (And there were also posts specifically about Banach-Mazur game and Choquet game.)

Apart from being a useful tool in some areas of research, they seem to be rather fascinating topic by itself, since there are some results which seem, at the first glance, rather counter-intuitive. (Especially when compared with finite games.) For example, it is possible that topological games is not determined, i.e., it is possible that neither of the two players has a winning strategy. (As far as I know, this is related to Axiom of Choice and Axiom of determinacy.) Another seemingly counter-intuitive fact is that for some games there might be difference between winning strategy and winning tactic (a.k.a. stationary strategy, which is a strategy where the move depends only on the position and not on the sequence of moves which lead to the position).

Is there some good text (book, thesis, lecture notes) suitable to start studying topological games and other similar infinite games?

Of course, I could start backwards by taking some paper using topological games in connection with some topic I am interested in or some survey article and whenever I stumble upon some unfamiliar result or notion, I could try to track down some references and learn about this particular thing. But if there is some reference which develops infinites game in a somewhat more systematic way, it might be more suitable way to start studying this topic.

(It is also possible that what I am asking is not a reasonable question in this sense: Maybe if somebody understands the basic idea of the topological games and winning strategies, then the next logical step to gain enough practice in adapting this technique to various specific situations rather than studying some systematic development of the topic.)


I suppose that for such specific topic there will not be too many suggestions. So I will be grateful for any suggestion. But since in the book recommendation and reference request it is advised to be specific, I will also add the following:

What would I mostly expect from the text? I would hope that after studying this text I would have better understanding of the seemingly counter-intuitive statements mentioned before. And I would also hope to be able to follow better proofs using topological games and maybe even to be able to come up with my own proofs in situations where topological games can be used.

If it is relevant to the question, I will also describe my background in this area. (Although I guess that if this post is supposed to be useful for other people as well, there should not be that much stress on this.) I understand the basic results about Banach-Mazur game and Choquet game given in Chapter 8 of Kechris: Classical Descriptive Set Theory. (I think I understand them well enough that probably I would be able to reproduce the proofs given enough time. I have not read Chapters 20 and 21, which also deal with topological games. I suppose that studying some material from the preceding parts of the book before starting these two chapters will be needed.) I have also been at some talks on topological games (in connection with Baire spaces), which I was more-or-less able to follow.

$\endgroup$
  • $\begingroup$ Actually, Chapters 20 & 21 make an excellent choice as a follow-up. Did you check Oxtoby's? It is cited in Kechris. $\endgroup$ – Pedro Sánchez Terraf Jan 21 '16 at 13:13
10
+50
$\begingroup$

Here are some suggestions. I'm afraid they do not fit exactly to your requirements; but I hope this will help anyway. I apologize in advance if you already know all of that.

0 Of course, Kechris and Oxtoby are great.

1 Concerning topological games like Banach-Mazur, and applications to Banach space theory, you could have a look at the following (and the references therein)

2 Concerning AD and these kinds of things:

3 Concerning the difference between strategies and tactics, try the following paper by Debs:

4 Incidentally, the best reference is perhaps Dens' "thèse d'état" entitled convexes compacts et jeux topologiques (but I didn't read it!)

5 A more specialized thing, which I suspect you may enjoy reading:

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.