text stringlengths 128 2.05k |
|---|
Proof of the main theorem. If [MATH] is a subset of [MATH] where [MATH] then for all [MATH] , we denote the successor of [MATH] in [MATH] by [MATH] |
Lemma 2.1 Let [MATH] be a non expanding set. Then there is [MATH] such that for all [MATH] , if [MATH] . Then, [MATH] Proof. For we may assume that [MATH] is infinite (Otherwise the Lemma is trivial). By contradiction assume that for all [MATH] , there is [MATH] such that [MATH] . Apply this assumption with [MATH] and ... |
Set [MATH] and [MATH] . For the rest of this paper, we will assume without loss of generality that [MATH] Let [MATH] . Let [MATH] |
Lemma 2.2 Let [MATH] . Let [MATH] . Then [MATH] if and only if for all [MATH] [MATH] iff [MATH] Proof. Immediate from the definition. |
Lemma 2.3 For all [MATH] for all [MATH] , if [MATH] then [MATH] Proof. By Lemma 2.2 , for all [MATH] [MATH] iff [MATH] . In particular, this is the case for all [MATH] . By Lemma 2.2 again, [MATH] |
Definition 2.4 [EQUATION] Lemma 2.5 For all [MATH] , if [MATH] then [MATH] Proof. Let [MATH] with [MATH] . Then by definition of [MATH] there is [MATH] such that [MATH] So by Lemma 2.3 |
[MATH] . Therefore by definition of [MATH] [MATH] Lemma 2.6 [MATH] Proof. First we remark that for all [MATH] [MATH] . Therefore by the Pigeonhole principle for all [MATH] , there is [MATH] such that [MATH] . Now by definition of [MATH] [MATH] |
Definition 2.7 [EQUATION] Lemma 2.8 For all [MATH] , there is [MATH] such that [MATH] Proof. By the Pigeonhole principle for all [MATH] , there is [MATH] such that [MATH] . Then by definition of [MATH] [MATH] . Take [MATH] |
The functions [MATH] come from Michaux-Villemaire . The authors prove that if [MATH] [MATH] . In fact, it is proved that for [MATH] large enough, [MATH] is defined by congruences relations modulo [MATH] . In our case this is not true anymore as [MATH] may not be in [MATH] . For instance, take [MATH] for some [MATH] . T... |
Definition 2.9 [EQUATION] Let [MATH] Lemma 2.10 [MATH] and [MATH] Proof. By Lemma 2.8 , for all [MATH] there is [MATH] such that [MATH] . Let [MATH] . This set is nonempty. For there is [MATH] such that [MATH] is minimal in [MATH] . Then as [MATH] [MATH] . By definition, [MATH] |
Let [MATH] . Set [EQUATION] Let [MATH] such that [MATH] . First if [MATH] then [MATH] (for either [MATH] and it is trivial as [MATH] or [MATH] and this follows from the definition of [MATH] ). Second if [MATH] then [MATH] for all [MATH] . So [MATH] . This proves that [MATH] This implies that [MATH] by definition of [MA... |
Let [MATH] such that [MATH] . Assume that [MATH] . As [MATH] [MATH] . As [MATH] we get that [MATH] . On the other hand, as [MATH] and [MATH] , we have that [MATH] . Now by definition of [MATH] [MATH] . We get a contradiction. Therefore [MATH] |
Definition 2.11 Let [MATH] . We say that [MATH] is cofinal in [MATH] if for all [MATH] there is [MATH] such that [MATH] Lemma 2.12 |
Let [MATH] definable such that [MATH] is cofinal in [MATH] and [MATH] then there is [MATH] definable such that [MATH] is cofinal in [MATH] and [MATH] is nondecreasing on [MATH] |
Proof. First, if [MATH] is finite: Then by the Pigeonhole principle there is [MATH] such that [MATH] is infinite. Set [MATH] . By definition of [MATH] [MATH] is constant on it. Also [MATH] is cofinal in [MATH] as it is infinite. Clearly [MATH] is definable. |
Second, if [MATH] is infinite. In that case, set [MATH] . By definition [MATH] is nondecreasing of [MATH] and [MATH] is definable. It remains to prove that [MATH] is an infinite set. We remark that [MATH] is non empty as [MATH] . Let [MATH] . Assume that for all [MATH] with [MATH] [MATH] . So, [MATH] is a finite set. T... |
Lemma 2.13 There is [MATH] definable such that [MATH] is cofinal in [MATH] and [MATH] are nondecreasing on [MATH] Proof. By Lemma 2.10 and Lemma 2.12 |
Lemma 2.14 [MATH] are definable maps. Proof. By definition, [MATH] are definable maps. By Lemma 2.13 [MATH] is definable. Lemma 2.15 |
Assume that [MATH] [MATH] and [MATH] are nonexpanding. Then, there is [MATH] , such that for all [MATH] , there is [MATH] such that [MATH] |
Proof. Let us remark that it is sufficient to prove that there is [MATH] , such that for all [MATH] , there is [MATH] such that [MATH] and [MATH] . For in that case, by Lemma 2.3 [MATH] |
By Lemma 2.1 and [MATH] is nondecreasing, there is [MATH] such that for all [MATH] [MATH] . Also there is [MATH] such that for all [MATH] [MATH] |
Claim 2.16 There are [MATH] such that for all [MATH] [MATH] , there are [MATH] and [MATH] with [MATH] Proof. First if [MATH] is eventually constant, take [MATH] [MATH] [MATH] |
and [MATH] . By definitions of [MATH] , we are done. Otherwise, [MATH] and [MATH] (as [MATH] is nondecreasing). Apply this with [MATH] . Then for all [MATH] there is [MATH] such that [MATH] . Take [MATH] . Then [MATH] . We remark that by definition of [MATH] |
[EQUATION] [EQUATION] We have that [MATH] . So, [EQUATION] Therefore by Lemma 2.3 , (1) and (2) [EQUATION] [EQUATION] We combine [MATH] and [MATH] to get |
[EQUATION] Take [MATH] and [MATH] . We remark that [MATH] , that [MATH] (by definition of [MATH] ) and that [MATH] (for remark that [MATH] and apply Lemma 2.3 and the above equality). Set [MATH] and [MATH] |
By the claim and the Pigeonhole principle there is [MATH] such that [MATH] is cofinal in [MATH] . Then, for all [MATH] there is [MATH] such that [MATH] . So, by the above claim, there is [MATH] such that [MATH] . This shows that there is [MATH] such that for all [MATH] there is [MATH] such that [MATH] and [MATH] . By t... |
Definition 2.17 Let [MATH] given by Lemma 2.15 . We define [EQUATION] By Lemma 2.15 , for all [MATH] [MATH] . Note that [MATH] is definable. |
Lemma 2.18 There is [MATH] definable such that [MATH] is cofinal in [MATH] and [MATH] is non decreasing on [MATH] Proof. By Lemma 2.12 |
From now, we will assume that [MATH] is restricted to [MATH] . We set [MATH] Lemma 2.19 If [MATH] [MATH] [MATH] and [MATH] are nonexpanding, then there is [MATH] and [MATH] such that [MATH] is a finite union of coset of [MATH] intersected with [MATH] (so is [MATH] -definable). |
Proof. By Lemma 2.1 , there is [MATH] such that for all [MATH] [MATH] . So for all [MATH] [MATH] Set [MATH] We have that for all [MATH] [MATH] , there is [MATH] [MATH] such that [MATH] . For there are two cases: first if there is [MATH] such that [MATH] . In that case [MATH] and we are done. Second if for all [MATH] |
[MATH] . In that case by Lemma 2.18 [MATH] is non decreasing. Furthermore, [MATH] (by Lemma 2.15 ). So as it is bounded by [MATH] [MATH] is eventually constant in [MATH] i.e., there is [MATH] such that for all [MATH] [MATH] [MATH] . Let [MATH] such that [MATH] (such [MATH] exists as [MATH] is cofinal in [MATH] see Lemm... |
Let [MATH] . By the above argument we know that there is [MATH] such that [MATH] . So by definition of [MATH] [MATH] . By Lemma 2.2 , this implies that [MATH] iff [MATH] . Therefore [MATH] . Take [MATH] be the set [MATH] . Then [MATH] |
Let [MATH] . This is a [MATH] -definable set. Therefore by DC there [MATH] such that [MATH] and either [MATH] or [MATH] . So we get that [MATH] and [MATH] is maximal for this property. As [MATH] [MATH] |
We can now prove the generalisation of the result of Michaux-Villemaire. Theorem 2.20 Let [MATH] and [MATH] . Assume that [MATH] admits EP and DC. Then for all [MATH] definable. [MATH] is a finite union of points and of cosets of [MATH] for some [MATH] |
Proof. By Proposition 1.5 [MATH] [MATH] [MATH] and [MATH] are nonexpanding. So the result is an immediate consequence of Lemma 2.19 |
This property is also true for [MATH] for all [MATH] Proposition 2.21 For all [MATH] there is [MATH] minimal, [MATH] and [MATH] maximal such that [MATH] is a finite union of classes of [MATH] intersected with [MATH] |
Proof. By Lemma 2.19 applied with the set [MATH] . For by Proposition 1.5 [MATH] [MATH] [MATH] and [MATH] are non expanding. Remark. We do not claim nor need that [MATH] or [MATH] are definable functions. |
Definition 2.22 [EQUATION] Lemma 2.23 Let [MATH] and [MATH] as given in Proposition 2.21 . Then for all [MATH] , if [MATH] is finite and [MATH] divides [MATH] then [MATH] . If [MATH] then [MATH] . Furthermore, for all [MATH] finite, [MATH] |
Proof. 1) First we prove that [MATH] : By Proposition 2.21 and definition of [MATH] [MATH] where [MATH] is a representative for the classe of [MATH] . As [MATH] , there is [MATH] maximal such that [MATH] . So, |
[EQUATION] Therefore by the above description of [MATH] [MATH] . So we remark that by definition of [MATH] (resp. [MATH] [EQUATION] |
[EQUATION] Also by definition of [MATH] [MATH] iff [MATH] for some [MATH] . This means that [MATH] . Similarly [MATH] This implies that [MATH] i.e., [MATH] . If [MATH] we are done by definition of [MATH] . Otherwise assume that [MATH] i.e., assume that there is [MATH] |
[MATH] . By Lemma 2.2 [MATH] iff [MATH] . So (as [MATH] divides [MATH] [MATH] . This contradicts the maximality of [MATH] in Proposition 2.21 . So [MATH] |
2) [MATH] : Assume that [MATH] . By definition of [MATH] [MATH] . By Lemma 2.3 [MATH] . So by Lemma 2.2 [MATH] iff [MATH] . Then by Proposition 2.21 |
[MATH] . This equality contradicts the minimality of [MATH] in Proposition 2.21 3) [MATH] : Assume [MATH] . By definition of [MATH] [MATH] . So by Lemma 2.3 |
[MATH] . By definition of [MATH] we get that [MATH] . On the other hand by Lemma 2.3 again [MATH] . Then by Lemma 2.2 , for all [MATH] [MATH] iff [MATH] . By Proposition 2.21 this proves that [MATH] divides [MATH] . Now as [MATH] , we get a contradiction with step 1). So, [MATH] |
Proposition 2.24 [MATH] is [MATH] -definable. Proof. By Proposition 2.21 , for all [MATH] , there is [MATH] and [MATH] maximal such that [MATH] is a union of classes of [MATH] restricted to [MATH] . So if there is [MATH] such that [MATH] we are done. Therefore for a contradiction we assume that for all [MATH] , for all... |
[MATH] For all [MATH] , there is [MATH] such that for all [MATH] [MATH] and furthermore, if [MATH] there is [MATH] such that for all [MATH] [MATH] Indeed, we can construct the [MATH] ’s by induction: Take [MATH] . Then, by Lemma 2.23 for all [MATH] [MATH] . For all [MATH] , set [MATH] . First remark that we may assume ... |
Claim 2.25 For all [MATH] [MATH] is not at finite distance from its predecessor in [MATH] (if any). Assume that the claim is true. Then we will build an elementary expansion of [MATH] such that [MATH] is an expanding set in this model (for some [MATH] ). Then we get a contradiction with the exchange property (by Propos... |
Let [MATH] be an ultrapower of [MATH] on a nonprincipal ultrafilter over [MATH] . Let [MATH] be the class of [MATH] and [MATH] be the class of [MATH] (the constant sequence). By construction of [MATH] and Łos Theorem, if [MATH] [MATH] . By the claim and Łos Theorem, for all [MATH] [MATH] i.e., for all [MATH] [MATH] is ... |
We give now a proof of the claim: By Lemma 2.23 as [MATH] divides [MATH] [MATH] . Let [MATH] such that [MATH] . We have to prove that the distance between these two elements is not finite. There are three possible cases: |
1) if [MATH] : in that case we may assume that [MATH] . For if [MATH] then [MATH] : contradiction with the choice of [MATH] . By definition of [MATH] [MATH] . So by Lemma 2.3 [MATH] . This proves that [MATH] : contradiction. So case 1) never occurs. |
2) [MATH] : Then by Lemma 2.23 [MATH] . As [MATH] [MATH] and [MATH] , we are done. 3) [MATH] : Assume that [MATH] is a finite distance from [MATH] . By Lemma 2.23 [MATH] where [MATH] are given by Proposition 2.21 . Then [MATH] is an infinite interval. Let [MATH] [MATH] . By Proposition 2.21 [MATH] for some [MATH] and [... |
# Source: arxiv 1806.00332 # Title: Every zero-dimensional homogeneous space is strongly homogeneous under determinacy # Sections: all # Downloaded: 2026-03-03T02:37:35.355629+00:00 |
Every zero-dimensional homogeneous space is strongly homogeneous under determinacy (Date: February 26, 2020) Abstract. All spaces are assumed to be separable and metrizable. We show that, assuming the Axiom of Determinacy, every zero-dimensional homogeneous space is strongly homogeneous (that is, all its non-empty clop... |
Key words and phrases: Homogeneous, strongly homogeneous, h-homogeneous, zero-dimensional, determinacy, Wadge theory, Hausdorff operation, [MATH] -ary Boolean operation. |
2010 Mathematics Subject Classification: 54H05, 03E15, 03E60. The first-listed author acknowledges the support of the FWF grant P 28153-N35. The second-listed author acknowledges the support of the FWF grant P 30823-N35. The third-listed author (formerly known as Sandra Uhlenbrock) acknowledges the support of the FWF g... |
1. Introduction Throughout this article, unless we specify otherwise, we will be working in the theory [MATH] , that is, the usual axioms of Zermelo-Fraenkel (without the Axiom of Choice) plus the principle of Dependent Choices (see Section 2 for more details). By space we will always mean separable metrizable topologi... |
A space [MATH] is strongly homogeneous (or h-homogeneous ) if every non-empty clopen subspace of [MATH] is homeomorphic to [MATH] . This notion has been studied by several authors, both “instrumentally” and for its own sake (see the list of references in Me1 ). It is well-known that every zero-dimensional strongly homo... |
Theorem 1.1 Assume [MATH] . If [MATH] is a zero-dimensional homogeneous space that is not locally compact then [MATH] is strongly homogeneous. |
The above theorem follows from a uniqueness result about zero-dimensional homogeneous spaces, namely Theorem 15.2 , which is of independent interest. This theorem essentially states that, for every sufficiently high level of complexity [MATH] , there are at most two homogeneous zero-dimensional spaces of complexity exa... |
Our fundamental tool will be Wadge theory, which was founded by William Wadge in his doctoral thesis Wa1 (see also Wa2 ), and has become a classical topic in descriptive set theory. We believe that vEMS , Theorem 2.4] and our results are the only applications to topology of an analysis of the full (as opposed to just B... |
[MATH] for some homogeneous [MATH] [MATH] is a good Wadge class [MATH] is closed under [MATH] and [MATH] [MATH] is reasonably closed Steel’s theorem can be applied to [MATH] |
The application of Wadge theory to the study of homogeneous spaces was pioneered by van Engelen in vE3 , where he obtained the classification mentioned above. As a corollary (namely, vE3 , Corollary 4.4.6] ), he obtained the Borel version of Theorem 1.1 . The reason why his results are limited to Borel spaces is that t... |
At this point, it is natural to ask whether assuming [MATH] is really necessary in the above results. As the following theorem shows, the answer is “yes”. This result was essentially proved in vD , but our exposition is based on vM , Theorem 5.1] . Following vM , we will say that [MATH] is a bi-Bernstein set if [MATH] ... |
Theorem 1.2 (van Douwen) There exists a [MATH] example [MATH] of a homogeneous zero-dimensional space that is not locally compact and not strongly homogeneous. |
Proof. Let [MATH] be the space given by vM , proof of Theorem 5.1] . Notice that [MATH] is homogeneous because [MATH] is a subgroup of [MATH] . Furthermore, [MATH] is a bi-Bernstein set by vM , Proposition 4.5] . It follows that both [MATH] and [MATH] are dense in [MATH] . In particular, [MATH] is zero-dimensional and ... |
Given any Borel subset [MATH] of [MATH] , pick a Borel subset [MATH] of [MATH] such that [MATH] , then define [MATH] , where [MATH] denotes the Lebesgue measure on [MATH] . Using the fact that [MATH] is bi-Bernstein, it is easy to check that [MATH] is a well-defined measure on the Borel subsets of [MATH] . The crucial ... |
Now pick [MATH] such that [MATH] . Observe that [MATH] and [MATH] are non-empty clopen subsets of [MATH] . Furthermore, it is clear from the definition of [MATH] that [MATH] . Therefore [MATH] and [MATH] are not homeomorphic, which concludes the proof. |
However, we do not know the answer to the following question. Recall that, when [MATH] or [MATH] for some [MATH] , a space is [MATH] if it is homeomorphic to a [MATH] subspace of some Polish space (see MZ , Section 4] for a more detailed treatment). |
Question 1.3 Assuming [MATH] , is it possible to construct a zero-dimensional [MATH] or [MATH] space that is homogeneous, not locally compact, and not strongly homogeneous? |
The above question is natural because there are many examples of properties (such as the perfect set property ) that are known to hold for all spaces under [MATH] , for which definable counterexamples can be constructed under [MATH] . Notice that [MATH] and [MATH] are optimal by vE3 , Corollary 4.4.6] . For other relev... |
Finally, we mention three applications of Theorem 1.1 . The first is that Theorem 1.2 cannot be proved without using the Axiom of Choice (more precisely, it cannot be proved in [MATH] alone). The second concerns the following question from Te , Section 3] (see Me1 , Section 3] and MvMZ , Section 5] for more on this top... |
Question 1.4 (Terada) Is [MATH] strongly homogeneous for every zero-dimensional first-countable space [MATH] Since Lawrence showed that [MATH] is homogeneous for every zero-dimensional space [MATH] (see La , or DP for a more general result), it follows from Theorem 1.1 that the answer to Question 1.4 in the separable m... |
Question 1.5 (Medvedev) Is every zero-dimensional meager homogeneous space strongly homogeneous? 2. Preliminaries and notation Let [MATH] be a set, and let [MATH] . Define [MATH] . We will say that [MATH] is selfdual if [MATH] . Also define [MATH] . Given a function [MATH] [MATH] , and [MATH] , we will use the notation... |
Definition 2.1 (Wadge) Let [MATH] be a space, and let [MATH] . We will write [MATH] if there exists a continuous function [MATH] such that [MATH] In this case, we will say that [MATH] is Wadge-reducible to [MATH] , and that [MATH] |
witnesses the reduction. We will write [MATH] if [MATH] and [MATH] . We will write [MATH] if [MATH] and [MATH] Definition 2.2 (Wadge) |
Let [MATH] be a space. Given [MATH] , define [EQUATION] We will say that [MATH] is a Wadge class if there exists [MATH] such that [MATH] . We will say that [MATH] is continuously closed if [MATH] for every [MATH] |
Both of the above definitions depend of course on the space [MATH] . Often, for the sake of clarity, we will specify what the ambient space is by saying, for example, that “ [MATH] in [MATH] ” or “ [MATH] is a Wadge class in [MATH] ”. We will say that [MATH] is selfdual if [MATH] in [MATH] . It is easy to check that [M... |
[MATH] [MATH] Our reference for descriptive set theory is Ke . In particular, we assume familiarity with the basic theory of Borel sets and Polish spaces, and use the same notation as in Ke , Section 11.B] . For example, given a space [MATH] , we use [MATH] [MATH] , and [MATH] to denote the collection of all open, clos... |
The classes defined below constitute the so-called difference hierarchy (or small Borel sets ). For a detailed treatment, see Ke , Section 22.E] or vE3 , Chapter 3] . Here, we will only mention that the [MATH] are among the simplest concrete examples of Wadge classes (see Proposition 9.3 and Corollary 9.5 ). |
Definition 2.3 (Kuratowski) Let [MATH] be a space, let [MATH] and [MATH] . Given a sequence of sets [MATH] , define [EQUATION] Define [MATH] if there exist [MATH] for [MATH] such that [MATH] |
For an introduction to the topic of games, we refer the reader to Ke , Section 20] . Here, we only want to give the precise definition of determinacy. A play of the game [MATH] is decribed by the diagram |
[MATH] [MATH] [MATH] II [MATH] [MATH] [MATH] in which [MATH] for every [MATH] and [MATH] is called the payoff set . We will say that Player I wins this play of the game [MATH] if [MATH] . Player II wins if Player I does not win. |
strategy for a player is a function [MATH] . We will say that [MATH] is a winning strategy for Player I if setting [MATH] for each [MATH] makes Player I win for every [MATH] . A winning strategy for Player II is defined similarly. We will say that the game [MATH] (or simply the set [MATH] ) is determined if (exactly) o... |
It is well-known that [MATH] is incompatible with the Axiom of Choice (see Je , Lemma 33.1] ). This is the reason why, throughout this article, we will be working in [MATH] Recall that the principle of Dependent Choices (briefly, [MATH] ) states that if [MATH] is a binary relation on a non-empty set [MATH] such that fo... |
We conclude this section with some miscellaneous topological definitions and results. We will write [MATH] to mean that the spaces [MATH] and [MATH] are homeomorphic. A subset of a space is clopen if it is closed and open. A space is zero-dimensional if it is non-empty and it has a base consisting of clopen sets. Given... |
Proposition 2.4 (Fitzpatrick, Zhou) Let [MATH] be a homogeneous space. Then [MATH] is either a meager space or a Baire space. Proposition 2.5 |
Let [MATH] be a zero-dimensional locally compact space. Then [MATH] is homogeneous iff [MATH] is discrete, [MATH] , or [MATH] Proof. |
The right-to-left implication is trivial. For the left-to-right implication, use the well-known characterization of [MATH] as the unique zero-dimensional crowded compact space (see Ke , Theorem 7.4] ). |
Proposition 2.6 Let [MATH] be a zero-dimensional homogeneous space. If there exists a non-empty Polish [MATH] then [MATH] is Polish. |
Proof. Let [MATH] be non-empty and Polish. Since [MATH] is zero-dimensional, we can assume without loss of generality that [MATH] . Let [MATH] . Notice that [MATH] is a cover of [MATH] because [MATH] is homogeneous and [MATH] is non-empty. Let [MATH] be a countable subcover of [MATH] . Define [MATH] for [MATH] , and ob... |
Proposition 2.7 Assume [MATH] . Let [MATH] be a Polish space, and let [MATH] be a dense Baire subspace of [MATH] . Then [MATH] is comeager in [MATH] |
Proof. Since [MATH] has the Baire property, we can write [MATH] by Ke , Proposition 8.23.ii] , where [MATH] and [MATH] is meager in [MATH] . It will be enough to show that [MATH] is dense in [MATH] . Assume, in order to get a contradiction, that there exists a non-empty open subset [MATH] of [MATH] such that [MATH] . O... |
Theorem 2.8 (Terada) Let [MATH] be a non-compact space. Assume that [MATH] has a base [MATH] such that [MATH] for every [MATH] . Then [MATH] is strongly homogeneous. |
3. The basics of Wadge theory The following simple lemma will allow us to generalize many Wadge-theoretic results from [MATH] to an arbitrary zero-dimensional Polish space. This approach has already appeared in An , Section 5] , where it is credited to Marcone. Recall that, given a space [MATH] and [MATH] , a retractio... |
Lemma 3.1 Let [MATH] , and let [MATH] be a retraction. Fix [MATH] . Then [MATH] in [MATH] iff [MATH] in [MATH] Proof. If [MATH] witnesses that [MATH] in [MATH] , then [MATH] will witness that [MATH] in [MATH] . On the other hand, if [MATH] witnesses that [MATH] in [MATH] , then [MATH] will witness that [MATH] in [MATH] |
The following result (commonly known as “Wadge’s Lemma”) shows that antichains with respect to [MATH] have size at most [MATH] Lemma 3.2 |
(Wadge) Assume [MATH] . Let [MATH] be a zero-dimensional Polish space, and let [MATH] . Then either [MATH] or [MATH] Proof. For the case [MATH] , see Ke , proof of Theorem 21.14] . To obtain the full result from this particular case, use Lemma 3.1 and the remarks preceding it. |
Theorem 3.3 (Martin, Monk) Assume [MATH] . Let [MATH] be a zero-dimensional Polish space. Then the relation [MATH] on [MATH] is well-founded. |
Proof. For the case [MATH] , see Ke , proof of Theorem 21.15] . To obtain the full result from this particular case, use Lemma 3.1 and the remarks preceding it. |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.