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Given a zero-dimensional Polish space [MATH] , define [EQUATION] Given [MATH] , define [MATH] if [MATH] for every [MATH] and [MATH] . Using the two previous results, one sees that the ordering [MATH] on [MATH] is a well-order. Therefore, there exists an order-isomorphism [MATH] for some ordinal [MATH] The reason for th... |
Definition 3.4 Let [MATH] be a zero-dimensional Polish space, and let [MATH] be a Wadge class in [MATH] . Define [EQUATION] We will say that [MATH] is the Wadge-rank of [MATH] |
It is easy to check that [MATH] is the minimal element of [MATH] . Furthermore, elements of the form [MATH] for [MATH] are always followed by [MATH] for some [MATH] , while elements of the form [MATH] for [MATH] are always followed by [MATH] for some [MATH] . This was proved by Van Wesep for [MATH] (see VW1 , Corollary... |
In fact, as Proposition 6.5 (together with Theorem 8.3 ) will show, the ordering of the non-selfdual classes is independent of the space [MATH] . However, the situation is more delicate for selfdual classes. For example, it follows easily from Corollary 4.2 that if [MATH] is a Wadge class in [MATH] such that [MATH] is ... |
The collection of all Wadge classes on a given space [MATH] , ordered by [MATH] , is known as the Wadge hierarchy . The following diagram shows how this hierarchy looks like when [MATH] is a zero-dimensional Polish space. |
[MATH] [MATH] [MATH] [MATH] [MATH] [MATH] We conclude this section with an elementary result, which shows that clopen sets are “neutral sets” for Wadge-reduction (the simple proof is left to the reader). By this we mean that, apart from trivial exceptions, intersections or unions with these sets do not change the Wadge... |
Proposition 3.5 Let [MATH] be a space, let [MATH] be a Wadge class in [MATH] , and let [MATH] Assume that [MATH] . Then [MATH] for every [MATH] |
Assume that [MATH] . Then [MATH] for every [MATH] 4. The analysis of selfdual sets In this section we will simply collect well-known results which show that every selfdual set can be built using non-selfdual sets of lower complexity (apply Corollary 4.2 with [MATH] ). We will refer to the proof of MR , Theorem 5.3] , w... |
Theorem 4.1 Assume [MATH] . Let [MATH] be a zero-dimensional Polish space, let [MATH] , and let [MATH] be a selfdual subset of [MATH] . Assume that [MATH] . Then there exist pairwise disjoint [MATH] for [MATH] such that [MATH] and [MATH] in [MATH] for each [MATH] |
Proof. This is proved like MR , Theorem 5.3] , with [MATH] instead of [MATH] (which is denoted by [MATH] there) and [MATH] , where [MATH] is the collection of all continuous [MATH] and [MATH] |
Corollary 4.2 Assume [MATH] . Let [MATH] be a zero-dimensional Polish space, let [MATH] , and let [MATH] be a selfdual subset of [MATH] . Then there exist pairwise disjoint [MATH] and non-selfdual [MATH] in [MATH] for [MATH] such that [MATH] and [MATH] |
Proof. As one can easily check, it will be enough to show that there exist pairwise disjoint [MATH] for [MATH] such that [MATH] and for every [MATH] either [MATH] or [MATH] is non-selfdual in [MATH] . If this were not the case, then, using Theorem 4.1 , one could recursively construct a strictly [MATH] -decreasing sequ... |
5. Basic facts on Hausdorff operations For a history of the following important notion, see Ha , page 583] . For a modern survey, we recommend Za . Most of the proofs in this section are straightforward, hence we leave them to the reader. |
Definition 5.1 Given a set [MATH] and [MATH] , define [EQUATION] whenever [MATH] . Functions of this form are called Hausdorff operations (or [MATH] -ary Boolean operations ). |
Of course, the function [MATH] depends on the set [MATH] , but what [MATH] is will usually be clear from the context. In case there might be uncertainty about the ambient space, we will use the notation [MATH] . Notice that, once [MATH] is specified, the corresponding Hausdorff operation simultaneously defines function... |
The following proposition lists the most basic properties of Hausdorff operations. Given [MATH] , define [MATH] Proposition 5.2 Let [MATH] be a non-empty set, and let [MATH] for every [MATH] . Fix an ambient set [MATH] and [MATH] |
[MATH] for all [MATH] [MATH] , where [MATH] [MATH] , where [MATH] [MATH] for all [MATH] The point of the above proposition is that any operation obtained by combining unions, intersections and complements can be expressed as a Hausdorff operation. For example, if [MATH] , then [MATH] |
The following proposition shows that the composition of Hausdorff operations is again a Hausdorff operation. We will assume that a bijection [MATH] has been fixed, and use the notation [MATH] |
Proposition 5.3 Let [MATH] be a set, let [MATH] and [MATH] for [MATH] . Then there exists [MATH] such that [EQUATION] for all [MATH] , where [MATH] |
Proof. Define [MATH] if [MATH] . The rest of the proof is a straightforward verification. We conclude this section with a result that will easily imply the fundamental Lemma 6.4 |
Proposition 5.4 Let [MATH] and [MATH] be sets, let [MATH] , let [MATH] and [MATH] (1) [MATH] whenever [MATH] (2) [MATH] for all [MATH] |
(3) [MATH] for all bijections [MATH] 6. Wadge classes and Hausdorff operations When one tries to give a systematic exposition of Wadge theory, it soon becomes apparent that it would be very useful to be able to talk about “abstract” Wadge classes, as opposed to Wadge classes in a particular space. More precisely, given... |
Definition 6.1 Given a space [MATH] and [MATH] , define [EQUATION] As examples (that will be useful later), consider the following two simple propositions. |
Proposition 6.2 Let [MATH] . Then there exists [MATH] such that [MATH] for every space [MATH] Proof. This follows from Propositions 5.2 and 5.3 (in case [MATH] , use a bijection [MATH] ). |
Proposition 6.3 Let [MATH] . Then there exists [MATH] such that [MATH] for every space [MATH] Proof. This can be proved by induction on [MATH] , using Propositions 5.2 and 5.3 |
Next, we obtain a very useful lemma, which shows that this notion behaves well with respect to subspaces and continuous functions. This lemma is essentially what we refer to when we speak about the “machinery of relativization”. It extends (and is inspired by) vE4 , Lemma 2.3] |
Lemma 6.4 Let [MATH] and [MATH] be spaces, and let [MATH] (1) Assume that [MATH] . Then [MATH] iff there exists [MATH] such that [MATH] |
(2) If [MATH] is continuous and [MATH] then [MATH] (3) If [MATH] is a homeomorphism then [MATH] iff [MATH] Proof. This is a straightforward consequence of Proposition 5.4 |
The following simple result, together with Theorem 8.3 , shows that the ordering of the non-selfdual Wadge classes is independent of the ambient space [MATH] (provided that [MATH] holds). |
Proposition 6.5 Let [MATH] and [MATH] be zero-dimensional spaces that contain a copy of [MATH] , and let [MATH] . Then [MATH] iff [MATH] |
Proof. Assume that [MATH] . Since [MATH] contains a copy of [MATH] and [MATH] is zero-dimensional, we see that [MATH] contains a copy of [MATH] . Using Lemma 6.4 , we can assume without loss of generality that [MATH] . Then |
[EQUATION] where the first and last equalities hold by Lemma 6.4 . The proof of the other implication is similar. 7. Universal sets |
The aim of this section is to prove the easier half of Theorem 8.3 (namely, Theorem 7.5 ). The ideas presented here are well-known, but since we could not find a satisfactory reference, we will give all the details. Our approach is inspired by Ke , Section 22.A] |
Definition 7.1 Let [MATH] and [MATH] be spaces, and let [MATH] . Given [MATH] and [MATH] , let [MATH] denote the vertical section of [MATH] above [MATH] . We will say that [MATH] is a [MATH] -universal set for [MATH] if the following two conditions hold: |
[MATH] [MATH] Notice that, by Proposition 6.3 , the above yields the definition of a [MATH] -universal set for [MATH] whenever [MATH] . Furthermore, this definition agrees with Ke , Definition 22.2] |
Proposition 7.2 Let [MATH] be a space, and let [MATH] . Then there exists a [MATH] -universal set for [MATH] Proof. By Ke , Theorem 22.3] , we can fix a [MATH] -universal set [MATH] for [MATH] . Let [MATH] be a homeomorphism, and let [MATH] be the projection on the [MATH] -th coordinate for [MATH] . Notice that, given ... |
We claim that [MATH] is a [MATH] -universal set for [MATH] . It is clear that [MATH] . Furthermore, using Lemma 6.4 , one can easily check that [MATH] for every [MATH] . To complete the proof, fix [MATH] . Let [MATH] be such that [MATH] . Since [MATH] is [MATH] -universal, we can fix [MATH] such that [MATH] for every [... |
Corollary 7.3 Let [MATH] be a space that contains a copy of [MATH] , and let [MATH] . Then there exists a [MATH] -universal set for [MATH] |
Proof. By Proposition 7.2 , we can fix a [MATH] -universal set [MATH] for [MATH] . Let [MATH] be such that [MATH] , and fix a homeomorphism [MATH] . Notice that [MATH] by Lemma 6.4 . Therefore, by Lemma 6.4 , there exists [MATH] such that [MATH] . Using Lemma 6.4 again, one can easily check that [MATH] is a [MATH] -uni... |
Lemma 7.4 Let [MATH] be a space, and let [MATH] . Assume that there exists a [MATH] -universal set for [MATH] . Then [MATH] is non-selfdual. |
Proof. Fix a [MATH] -universal set [MATH] for [MATH] . Assume, in order to get a contradiction, that [MATH] is selfdual. Let [MATH] be the function defined by [MATH] , and observe that [MATH] is continuous. Since [MATH] , we see that [MATH] . Therefore, since [MATH] is [MATH] -universal, we can fix [MATH] such that [MA... |
The case [MATH] of the following result is VW1 , Proposition 5.0.3] , and it is credited to Addison by Van Wesep. Theorem 7.5 Let [MATH] be a zero-dimensional space that contains a copy of [MATH] , and let [MATH] . Then [MATH] |
Proof. The fact that [MATH] is non-selfdual follows from Corollary 7.3 and Lemma 7.4 . Therefore, it will be enough to show that [MATH] is a Wadge class. By Proposition 7.2 , we can fix a [MATH] -universal set [MATH] for [MATH] . Let [MATH] be such that [MATH] , and fix a homeomorphism [MATH] . By Lemma 6.4 , we can fi... |
8. Van Wesep’s theorem The following is one of the main results of Van Wesep’s doctoral thesis (see VW1 , Theorem 5.3.1] , whose proof also employs results of Steel from St1 and results of Radin), and it will allow us to obtain the harder half of Theorem 8.3 . Notice how Corollary 8.2 guarantees that every non-selfdual... |
Theorem 8.1 (Van Wesep) Assume [MATH] . For every [MATH] there exists [MATH] such that [MATH] Corollary 8.2 Assume [MATH] . Let [MATH] be a zero-dimensional Polish space, and let [MATH] . Then there exists [MATH] such that [MATH] |
Proof. By Ke , Theorem 7.8] , there exists a closed [MATH] such that [MATH] . Therefore, using Lemma 6.4 , we can assume without loss of generality that [MATH] is a closed subspace of [MATH] . Hence, by Ke , Proposition 2.8] , we can fix a retraction [MATH] . Let [MATH] be such that [MATH] . Set [MATH] , and let [MATH]... |
Using Lemma 3.1 , it is easy to see that [MATH] . Therefore, by Theorem 8.1 , we can fix [MATH] such that [MATH] . We claim that [MATH] . Notice that [MATH] by Lemma 6.4 , hence [MATH] by Lemma 6.4 . Finally, to see that [MATH] , pick [MATH] . Observe that [MATH] by Lemma 6.4 . This means that [MATH] in [MATH] , hence ... |
Finally, we can “put everything together” and state the full result promised in the introduction to Section 6. Theorem 8.3 Assume [MATH] . Let [MATH] be an uncountable zero-dimensional Polish space. Then |
[EQUATION] Proof. This follows immediately from Theorem 7.5 and Corollary 8.2 9. Basic facts on expansions The following notion is essentially due to Wadge (see Wa1 , Chapter IV] ), and it is inspired by work of Kuratowski. Recall that, given [MATH] and spaces [MATH] and [MATH] , a function [MATH] is [MATH] -measurable... |
Definition 9.1 Let [MATH] be a space, and let [MATH] . Given [MATH] , define [EQUATION] We will refer to [MATH] as an expansion of [MATH] |
The following is the corresponding definition in the context of Hausdorff operations. Corollary 10.4 below shows that this is in fact the “right” definition. |
Definition 9.2 Let [MATH] be a space, let [MATH] , and let [MATH] . Define [EQUATION] As an example (that will be useful later), consider the following simple observation. |
Proposition 9.3 Let [MATH] . Then there exists [MATH] such that [MATH] for every space [MATH] and every [MATH] Proof. This is proved like Proposition 6.2 (in fact, the same [MATH] will work). |
The following proposition shows that Definition 9.2 actually fits in the context provided by Section 6. Proposition 9.4 Let [MATH] , and let [MATH] . Then there exists [MATH] such that [MATH] for every space [MATH] |
Proof. This is proved by combining Propositions 6.3 and 5.3 Corollary 9.5 Let [MATH] be an uncountable zero-dimensional Polish space, let [MATH] , and let [MATH] . Then [MATH] |
Proof. This is proved by combining Proposition 9.4 and Theorem 7.5 The following useful result is the analogue of Lemma 6.4 in the present context. |
Lemma 9.6 Let [MATH] and [MATH] be spaces, let [MATH] , and let [MATH] (1) Assume that [MATH] . Then [MATH] iff there exists [MATH] such that [MATH] |
(2) If [MATH] is continuous and [MATH] then [MATH] (3) If [MATH] is [MATH] -measurable and [MATH] then [MATH] (4) If [MATH] is a homeomorphism then [MATH] iff [MATH] |
Proof. This is a straightforward consequence of Proposition 5.4 10. Kuratowski’s transfer theorem The aim of this section is to collect the tools needed to successfully employ the notion of expansion. For example, Corollary 10.3 will be a crucial ingredient in the proof of Theorem 11.3 . A stronger form of Theorem 10.1... |
Theorem 10.1 (Kuratowski) Let [MATH] be a Polish space, let [MATH] , and let [MATH] be countable. Then there exists a zero-dimensional Polish topology [MATH] on the set [MATH] such that [MATH] and [MATH] |
Proof. This follows from Ke , Exercise 22.20] , using the fact that every element of [MATH] can be written as a countable union of elements of [MATH] |
Corollary 10.2 Let [MATH] be a zero-dimensional Polish space, let [MATH] , and let [MATH] be countable. Then there exists a zero-dimensional Polish space [MATH] and a [MATH] -measurable bijection [MATH] such that [MATH] for every [MATH] |
Proof. The case [MATH] is trivial, so assume that [MATH] . The space [MATH] is simply the set [MATH] with the finer topology given by Theorem 10.1 , while [MATH] |
Corollary 10.3 Let [MATH] be a zero-dimensional Polish space, let [MATH] , and let [MATH] . Assume that [MATH] and [MATH] are countable. Then there exists a zero-dimensional Polish space [MATH] and a [MATH] -measurable bijection [MATH] such that [MATH] for every [MATH] and [MATH] for every [MATH] |
Proof. Let [MATH] . Given [MATH] , fix [MATH] for [MATH] such that [MATH] . Define [MATH] . By Corollary 10.2 , we can fix a Polish space [MATH] and a [MATH] -measurable bijection [MATH] such that [MATH] for every [MATH] . It remains to observe that |
[EQUATION] for every [MATH] , where the second equality follows from Proposition 5.4 Corollary 10.4 Let [MATH] be an uncountable zero-dimensional Polish space, let [MATH] , and let [MATH] . Then [MATH] |
Proof. The inclusion [MATH] follows from Lemma 9.6 . In order to prove the other inclusion, pick [MATH] . By Corollary 10.3 , we can fix a zero-dimensional Polish space [MATH] and a [MATH] -measurable bijection [MATH] such that [MATH] . Since [MATH] contains a copy of [MATH] and [MATH] is zero-dimensional, using Lemma ... |
Corollary 10.5 Assume [MATH] . Let [MATH] be an uncountable zero-dimensional Polish space, and let [MATH] . Then [MATH] for every [MATH] |
Proof. This follows from Corollary 8.2 , Corollary 10.4 , and Proposition 9.5 Corollary 10.6 Assume [MATH] . Let [MATH] be an uncountable zero-dimensional Polish space, and let [MATH] . Then [MATH] iff [MATH] for every [MATH] |
Proof. The fact that [MATH] implies [MATH] is a trivial consequence of the definition of expansion. Now fix [MATH] such that [MATH] . Assume, in order to get a contradiction, that [MATH] . Then [MATH] by Lemma 3.2 , hence |
[EQUATION] Since [MATH] is non-selfdual by Corollary 10.5 , this is a contradiction. 11. The expansion theorem The main result of this section is Theorem 11.3 , which will be a crucial tool in obtaining the closure properties in the next section, and will be referred to as the expansion theorem. The proof given here is... |
Definition 11.1 (Wadge) Let [MATH] be a space, let [MATH] , and let [MATH] . Define [MATH] to be the collection of all sets of the form |
[EQUATION] where each [MATH] , each [MATH] , the [MATH] are pairwise disjoint, and [MATH] . A set in this form is called a partitioned union of sets in [MATH] |
Notice that the sets [MATH] in the above definition are not required to be non-empty. It is easy to check that [MATH] is continuously closed whenever [MATH] is. Furthermore, it is clear that |
[EQUATION] whenever [MATH] and [MATH] Definition 11.2 (Louveau, Saint-Raymond) Let [MATH] be a space, let [MATH] be continuously closed, and let [MATH] . Define |
[MATH] if [MATH] [MATH] if [MATH] and [MATH] [MATH] if [MATH] for every [MATH] We refer to [MATH] as the level of [MATH] As a trivial example, observe that [MATH] . Using the definition of Wadge-reduction, it is a simple exercise to see that [MATH] for every Wadge class [MATH] . We remark that it is not clear at this p... |
Theorem 11.3 Assume [MATH] . Let [MATH] be an uncountable zero-dimensional Polish space, let [MATH] , and let [MATH] . Then the following conditions are equivalent: |
(1) [MATH] (2) [MATH] for some [MATH] Proof. In order to show that [MATH] , assume that [MATH] . Let [MATH] be minimal with respect to the property that [MATH] . Assume, in order to get a contradiction, that [MATH] . It follows from Lemma 3.2 that [MATH] , hence [MATH] . Fix [MATH] such that [MATH] . Also fix [MATH] su... |
Next, we will show that [MATH] . Assume, in order to get a contradiction, that this is not the case. By Corollary 8.2 , we can fix [MATH] such that [MATH] . Notice that [MATH] . Furthermore [MATH] , hence [MATH] . Since [MATH] is non-selfdual by Theorem 7.5 , it follows that [MATH] . Therefore, [MATH] by Proposition 6.... |
Since [MATH] , by Corollaries 4.2 and 8.2 , we can fix [MATH] [MATH] and pairwise disjoint [MATH] for [MATH] such that [MATH] and [MATH] for each [MATH] . Notice that [MATH] for each [MATH] by Proposition 6.5 , hence [MATH] for each [MATH] by the minimality of [MATH] . It follows from Corollary 10.4 and Lemma 3.2 that ... |
Set [MATH] for [MATH] . Observe that [MATH] for each [MATH] by Lemma 9.6 . Furthermore, it is clear that [MATH] for each [MATH] . In conclusion, since [MATH] , we see that |
[EQUATION] where the last equality uses the assumption that [MATH] . This contradicts the fact that [MATH] is non-selfdual. In order to show that [MATH] , let [MATH] be such that [MATH] . Pick [MATH] and pairwise disjoint [MATH] for [MATH] such that [MATH] . We need to show that [MATH] . By Corollary 8.2 , we can fix [... |
[EQUATION] where the second equality holds by Corollary 10.4 Corollary 11.4 Assume [MATH] . Let [MATH] and [MATH] be uncountable zero-dimensional Polish spaces, let [MATH] , and let [MATH] . Then [MATH] iff [MATH] |
Proof. We will only prove the left-to-right implication, as the other one can be proved similarly. Assume that [MATH] . Then, by Theorems 7.5 and 11.3 , there exists [MATH] such that [MATH] . By Corollary 8.2 , we can fix [MATH] such that [MATH] . By Proposition 9.4 , we can fix [MATH] such that [MATH] and [MATH] . Not... |
12. Good Wadge classes The following key notion is essentially due to van Engelen, although he did not give it a name. One important difference is that van Engelen’s treatment of this notion is fundamentally tied to Louveau’s classification of the Borel Wadge classes from Lo1 , hence it is limited to the Borel context.... |
Definition 12.1 Let [MATH] be a space, and let [MATH] be a Wadge class in [MATH] . We will say that [MATH] is good if the following conditions are satisfied: |
[MATH] is non-selfdual, [MATH] [MATH] The following proposition gives some concrete examples of good Wadge classes. Proposition 12.2 |
Let [MATH] be an uncountable zero-dimensional Polish space, let [MATH] , and let [MATH] . Then [MATH] is a good Wadge class in [MATH] |
Proof. Set [MATH] . The fact that [MATH] follows from Propositions 9.3 and 9.5 . The inclusion [MATH] holds trivially. Finally, using Corollary 10.4 and LSR1 , Théorème 8] one sees that [MATH] |
The main result of this section is Theorem 12.4 , which will be crucial in showing that good Wadge classes are reasonably closed (see Lemma 13.2 ). The case [MATH] of the following lemma is due to Andretta, Hjorth, and Neeman (see AHN , Lemma 3.6.a] ), and the general case follows easily from this particular case (than... |
Lemma 12.3 Assume [MATH] . Let [MATH] be an uncountable zero-dimensional Polish space, and let [MATH] . Assume that [MATH] for every [MATH] |
If [MATH] and [MATH] then [MATH] If [MATH] and [MATH] then [MATH] Proof. Observe that, since [MATH] also satisfies the assumptions of the lemma, it will be enough to prove the first statement. So pick [MATH] and [MATH] . By Corollary 8.2 , we can fix [MATH] such that [MATH] . Set [MATH] . Using Lemma 6.4 , we can assum... |
Next, we claim that [MATH] (see Definition 3.4 ). Since [MATH] for every [MATH] , using Propositions 6.2 and 6.5 one sees that [MATH] for every [MATH] . Since these are Wadge classes by Theorem 7.5 , and they form a strictly increasing sequence by Ke , Exercise 22.26.iv] , our claim is proved. Therefore, we can apply A... |
Theorem 12.4 Assume [MATH] . Let [MATH] be an uncountable zero-dimensional Polish space, and let [MATH] . Assume that [MATH] for every [MATH] and [MATH] |
If [MATH] and [MATH] then [MATH] If [MATH] and [MATH] then [MATH] In particular, the above two statements hold for every good Wadge class [MATH] in [MATH] |
Proof. Observe that, since [MATH] also satisfies the assumptions of the theorem, it will be enough to prove the first statement. So pick [MATH] and [MATH] . By Theorem 11.3 , we can pick [MATH] such that [MATH] . By Corollary 8.2 , we can fix [MATH] such that [MATH] |
Since [MATH] , there exists a [MATH] -measurable function [MATH] and [MATH] such that [MATH] . Furthermore, using Corollary 10.4 for a suitable choice of [MATH] , it is easy to check that [MATH] . Therefore, there exists a [MATH] -measurable function [MATH] and [MATH] such that [MATH] . By applying Lemma 6.4 to the pro... |
We claim that [MATH] for every [MATH] . So fix [MATH] , and let [MATH] be the set given by Proposition 9.3 when [MATH] . Notice that |
[EQUATION] where the first equality holds by Corollary 10.4 . Therefore [MATH] by Corollary 10.6 . An application of Proposition 6.5 with [MATH] concludes the proof of our claim. |
Therefore, we can apply Lemma 12.3 , which shows that [MATH] . Consider the function [MATH] defined by [MATH] , and observe that [MATH] is [MATH] -measurable. By Lemma 9.6 , it follows that |
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