Stationary strategies in Choquet games
The (strong) Choquet game on a topological space is played as follows. There are two players, Empty and Nonempty, who alternate turns for infinitely many rounds. On round , Empty moves first, choosing a point and an open neighborhood of and, if , such that (the open set that Nonempty played on the previous round). Then, Nonempty responds with an open neighborhood of the same point such that . After all the rounds have been played, we obtain a descending sequence of open sets
together with a sequence of points Empty wins this play if ; Nonempty wins if .
A Choquet space is a topological space such that Nonempty has a winning strategy in the Choquet game played on the topological space . The Choquet game was originally designed by Choquet to give a topological characterization of which metrizable spaces admit a complete metric. However, not all Choquet spaces are metrizable. In general, the Choquet game turns out to be a good measure of completeness for topological spaces.
In the case of complete metric spaces , Nonempty has a relatively simple winning strategy in the Choquet game on . Once Empty has played the point-neighborhood pair , Nonempty responds by picking an open ball around that fits inside and has radius no larger than . This forces Empty to play a Cauchy sequence of points whose limit witnesses that . Note that to carry out this strategy, Nonempty only needs to know the last move played by Empty and to remember which round is currently being played. In fact, with just a small change in strategy, Nonempty doesn’t even need to remember which round is being played: Nonempty simply needs to pick an open ball around whose radius is no larger than a quarter of the diameter of since that ensures that the radius of each open ball played by Nonempty decreases by at least one half at each step.
A strategy for Nonempty that only uses the last move played by Empty to decide what to play next is called a stationary strategy. Thus, we see that for a metrizable space , the following are equivalent:
- admits a complete metric.
- Nonempty has a winning strategy in the Choquet game played on .
- Nonempty has a stationary winning strategy in the Choquet game played on .
Since the Choquet game makes sense for arbitrary topological spaces, it makes sense to ask whether items 2 and 3 are equivalent in the general case. This is not the case, but it is known that the equivalence holds for classes of spaces much broader than metrizable spaces.
In our paper (Dorais–Mummert 2010), Carl Mummert and I show that the equivalence between the existence of general and stationary strategies for Nonempty in the Choquet game holds for an interesting class of spaces, which includes all second-countable T1 spaces. To state our main result, I must introduce an unusual property of topological bases. A base for a topology is said to be open-finite if every open set has only finitely many supersets in . While it is unusual for a base to have this property, it turns out that many spaces happen to have such a base. For example, all second-countable T1 spaces have such a base. The main result of our paper is the following.
Theorem (Dorais–Mummert 2010). Let be a topological space with an open-finite base. If Nonempty has a winning strategy in the Choquet game on , then Nonempty has a stationary winning strategy in the Choquet game on .
The method for proving this is new and interesting, but you will have to read our paper to find out…
The Choquet game appears to be tied with certain types of representability of topological spaces. Representability issues are very important in the context of reverse mathematics since second-order arithmetic offers very limited resources to talk about large multi-layered objects like topological spaces. In (Mummert 2006), Carl Mummert introduced a broad class of topological spaces that can be represented in second-order arithmetic: countably based maximal filter (MF) spaces. The basic datum for these spaces consists of a countable partial order , the points of the space is the class of maximal filters on , and the basic open sets consist of all classes . It is not hard to see that these second-countable spaces are all T1 and Choquet.
A topological characterization of countably based MF spaces was obtained by Mummert and Stephan (2010), who established that the countably based MF spaces are precisely the second-countable T1 Choquet spaces. The original proof of this result is long and intricate. The existence of stationary winning strategies for Nonempty in such spaces leads to a much easier proof of this representation theorem. This proof was not included in our paper since it was too far from the main topic and not short enough to include in passing. Therefore, I am recording this proof here for prosperity.
Theorem (Mummert–Stephan 2010). Every second-countable T1 Choquet space is homeomorphic to a countably based MF space.
Proof. Suppose that is a second-countable T1 Choquet space. Let be a countable open-finite base for , and let be a stationary strategy for Nonempty in the Choquet game on . We will define a transitive relation on such that is homeomorphic to . (Although the notation suggests otherwise, the relation is not necessarily irreflexive.)
A natural choice for would be to define to hold if and only if for some . However, this relation is not necessarily transitive. To remedy this, we define to hold if and only if there is a point such that for every such that . This relation is clearly transitive. Moreover, since is open-finite, there are only finitely many such , so the intersection of all corresponding is an open neighborhood of . This guarantees that for every , there is a such that and .
We begin by recording a lemma that will be used repeatedly in this proof.
Lemma. For every maximal filter on there is a descending sequence
such that .
Proof. Since is countable and downward directed in , it is easy to get a sequence
such that . If this sequence is not eventually constant, we can eliminate repeated elements to obtain the a sequence as required by the lemma. Otherwise, we may assume that for every . As observed above, there must be some such that . Since is a maximal filter in , we must have , which means that the constant sequence is as required by the lemma. QED
The first step of the proof is to define the map that will witness that the two spaces are homeomorphic. Fix , we will show that is always a singleton, so that we may define to be the unique point of that belongs to every element of .
First find a descending sequence
that generates as in the above lemma. By definition of , we can find corresponding points such that and . This defines a valid sequence of moves for Empty against Nonempty’s stationary strategy in the Choquet game on . Since is a winning strategy for Nonempty, it follows that is nonempty.
To see that has only one point, suppose for the sake of contradiction that contains two distinct points and . Because is T1, we can find a neighborhood of in that does not contain . Define the descending sequence
so that and for every such that . The filter
extends since for each . Since but , this contradicts the maximality of .
Now that is properly defined, it remains to show that it is a homeomorhism. We first show that is a bijection, which we break into two facts:
- is injective. Suppose that and are maximal filters that map to the same point . By the lemma, we can find two sequences
that generate these two filters. Since for each , we can find another sequence
of neighborhoods of in such that for every such that . Then the filter
extends both and , which means that .
- is surjective. Let be an enumeration of (possibly with repetitions). Given , define the descending sequence
of neighborhoods of in as follows. Pick so that . If then pick in such a way that for every such that ; if then simply set . Since is T1, we immediately see that . Therefore, any maximal filter extending
will map to . (In fact, is already maximal.) Since the choice of was essentially arbitrary in the process we just used to show that is surjective, for every and every we can find some such that and . It follows that
which shows that is a homeomorphism. QED
F. G. Dorais, C. Mummert, 2010: Stationary and convergent strategies in Choquet games, Fund. Math. 209, 59–79.
C. Mummert, 2006: Reverse mathematics of MF spaces, J. Math. Log. 6, 203–232.
C. Mummert, F. Stephan, 2010: Topological aspects of poset spaces, Michigan Math. J. 59, 3–24.