The classical approach initiated by the French analysts Baire, Borel, and Le-besgue to the classification of the subsets of a metric space is more descriptive in nature. Sets are classified according to the complexity of their definition from open sets. This approach was continued later by Luzin, Suslin, Haus-dorff, Sierpiński and Kuratowski.
As observed – apparently for the first time – by Tang [Tan79; Tan81], the classical definition of the Borel hierarchy in metric spaces is not satisfactory for non metrisable spaces. Following Selivanov [Sel06] and de Brecht [deB13]
we use the following slightly modified definition of the Borel hierarchy in an arbitrary topological space (see also the paper by Spurný [Spu10]).
Definition 5.39. Let 𝒳 be a topological space. For each positive ordinal 𝛼 < 𝜔1 we define by induction
Proposition 5.40. For any topological space 𝒳 and any 𝛼 > 0:
(i) 𝚺0𝛼(𝒳) is closed under countable union and finite intersection;
(ii) 𝚷0𝛼(𝒳) is closed under countable intersection and finite union;
(iii) 𝚫0𝛼(𝒳) is closed under finite union and intersection as well as under complementation.
Proposition 5.42. If 𝛼 > 2, then 𝚺0𝛼(𝒳) = {⋃
𝑖∈𝜔
𝐵𝑖∣ 𝐵𝑖 ∈ ⋃
𝛽<𝛼
𝚷0𝛽(𝒳) for each 𝑖 ∈ 𝜔}.
And if 𝒳 is metrisable the previous statement holds also for 𝛼 = 2, i.e.
𝚺02(𝒳) = {⋃
𝑖∈𝜔
𝐵𝑖 ∣ 𝐵𝑖∈ 𝚷01(𝒳) for each 𝑖 ∈ 𝜔}.
Hausdorff and later Kuratowski refined the Borel hierarchy by introducing the so called difference Hierarchy. Recall that any ordinal 𝛼 can uniquely be expressed as𝛼 = 𝜆 + 𝑛 where 𝜆 is limit or equal to0, and𝑛 < 𝜔. The ordinal 𝛼 is said to be even if 𝑛is even, otherwise 𝛼 is said to be odd.
Definition 5.43. Let𝜉 ⩾ 1 be a countable ordinal. For any sequence(𝐶𝜂)𝜂<𝜉 with 𝛼 < 𝛽 < 𝜉 implies 𝐶𝛼 ⊆ 𝐶𝛽, the set𝐴 = 𝐷𝜉((𝐶𝜂)𝜂<𝜉) is defined by
𝐴 = {⋃{𝐶𝜂∖ ⋃𝜂′<𝜂𝐶𝜂′ ∣ 𝜂 odd, 𝜂 < 𝜉} for 𝜉 even,
⋃{𝐶𝜂∖ ⋃𝜂′<𝜂𝐶𝜂′ ∣ 𝜂 even, 𝜂 < 𝜉} for 𝜉 odd.
For a topological space 𝒳, 1 ⩽ 𝛼 < 𝜔1 and 1 ⩽ 𝜉 < 𝜔1 we let 𝐷𝜉(𝚺0𝛼(𝒳)) be the class of all sets𝐷𝜉((𝐶𝜂)𝜂<𝜉)where(𝐶𝜂)𝜂<𝜉 is an increasing sequence in 𝚺0𝛼(𝒳). Notice that in particular 𝐷1(𝚺0𝛼) = 𝚺0𝛼.
Of course if 𝑓 ∶ 𝒳 → 𝒴 is a continuous map and 𝐴 ∈ 𝐷𝜉(𝚺0𝛼(𝒴)), then 𝑓−1(𝐴) ∈ 𝐷𝜉(𝚺0𝛼(𝒳)). For the Baire space, this straightforward observation is crystallised in the definition of apointclass, that is a collection of subsets of the Baire space closed under continuous preimages, or in other words, an initial segment of the Wadge quasi-order on the Baire space. In any topological space, the fact that the classes 𝚷0𝛼, 𝚺0𝛼 and 𝐷𝜉(𝚺0𝛼) are closed under continuous preimages means that these classes are initial segment of the quasi-order of continuous reducibility⩽W. In this sense⩽Wrefines these classical hierarchies.
We now show that in an arbitrary second countable 𝑇0 space 𝒳 the classes 𝚷0𝛼,𝚺0𝛼,𝐷𝜉(𝚺0𝛼)enjoy the stronger and less straightforward property of being initial segments of the quasi-order ≼W. Therefore the quasi-order ≼W also refines these classical hierarchies.
Proposition 5.44. Let 𝒳 and 𝒴 be second countable 𝑇0 spaces and 𝐴 ⊆ 𝒳, 𝐵 ⊆ 𝒴. For every 1 ⩽ 𝛼, 𝜉 < 𝜔1,
(i) if 𝐵 ∈ 𝚺0𝛼(𝒴) and 𝐴 ≼W 𝐵, then 𝐴 ∈ 𝚺0𝛼(𝒳),
(ii) if 𝐵 ∈ 𝐷𝜉(𝚺0𝛼(𝒴)) and 𝐴 ≼W 𝐵, then 𝐴 ∈ 𝐷𝜉(𝚺0𝛼(𝒳)).
We defer the proof of the previous proposition until the end of this section since it follows from results of independent interest. The proof relies essentially on the following proposition which is a slightly modified version of a result due to Saint Raymond [Sai07, Lemma 17]. Its relevance in our context was first observed by de Brecht [deB13]. It is based on Baire category and we refer the reader to the textbook by Kechris [Kec95] for the basic definitions and results.
Proposition 5.45. Let 𝒳 and𝒴 be topological spaces, with 𝒳metrisable. Let 𝜑 ∶ 𝒳 → 𝒴 be an open, continuous map with Polish fibres, i.e.𝜑−1(𝑦)is Polish for all 𝑦 ∈ 𝒴. For every 𝑍 ⊆ 𝒳 define:
𝑁0(𝑍) = {𝑦 ∈ 𝒴 ∣ 𝑍 ∩ 𝜑−1(𝑦) is non meagre in 𝜑−1(𝑦)}, 𝑁1(𝑍) = {𝑦 ∈ 𝒴 ∣ 𝑍 ∩ 𝜑−1(𝑦) is comeagre in 𝜑−1(𝑦)}.
Then for every positive ordinal 𝛼 < 𝜔1, (i) If 𝑍 ∈ 𝚺0𝛼(𝒳), then 𝑁0(𝑍) ∈ 𝚺0𝛼(𝒴), (ii) If 𝑍 ∈ 𝚷0𝛼(𝒳), then 𝑁1(𝑍) ∈ 𝚷0𝛼(𝒴).
In particular, if𝜑is further assumed to be surjective then for every𝐴 ⊆ 𝒴and every positive ordinal 𝛼 < 𝜔1,
(i) 𝜑−1(𝐴) ∈ 𝚺0𝛼(𝒳) ⟷ 𝐴 ∈ 𝚺0𝛼(𝒴), (ii) 𝜑−1(𝐴) ∈ 𝚷0𝛼(𝒳) ⟷ 𝐴 ∈ 𝚷0𝛼(𝒴).
Proof. Since𝑁1(𝒳 ∖ 𝑍) = 𝒴 ∖ 𝑁0(𝑍) both statements are equivalent for every 𝛼. Let (𝑉𝑘)𝑘∈𝜔 be a countable base for the topology of 𝒳. We proceed by induction on𝛼.
For𝛼 = 1 let𝑍 ∈ 𝚺01, since 𝜑 is assumed to be open we have 𝜑(𝑍) is open in𝒴. Since𝜑−1(𝑦) is a Baire space for all𝑦 ∈ 𝒴, the open subset𝑍 ∩ 𝜑−1(𝑦)of 𝜑−1(𝑦)is non meagre if and only if it is non empty. So𝑁0(𝑍) = 𝜑(𝑍) ∈ 𝚺01(𝒴).
So assume now that both statements are true for every 𝛼′ < 𝛼 and let 𝑍 ∈ 𝚺0𝛼. Since 𝒳is metrisable, 𝑍 is the union of a countable family(𝑍𝑛)𝑛∈𝜔 with𝑍𝑛∈ 𝚷0𝛼
𝑛 for some 𝛼𝑛 < 𝛼. For any point𝑦 ∈ 𝒴, using the fact that any Borel subset of a Polish space has the Baire Property, we have the following equivalences:
𝑍 ∩ 𝜑−1(𝑦) is non meagre in 𝜑−1(𝑦)
↔ ∃𝑛 𝑍𝑛∩ 𝜑−1(𝑦) is non meagre in 𝜑−1(𝑦)
↔ ∃𝑛 𝑍𝑛∩ 𝜑−1(𝑦) is comeagre in some non-empty open subset of 𝜑−1(𝑦)
↔ ∃𝑛 ∃𝑘 (𝑍𝑛∪ 𝑉𝑘∁) ∩ 𝜑−1(𝑦) is comeagre in 𝜑−1(𝑦) and 𝑉𝑘∩ 𝜑−1(𝑦) ≠ ∅.
Therefore,
For the second claim, assume that 𝜑is moreover surjective and notice that if 𝐴 ⊆ 𝒴then for 𝑍 = 𝜑−1(𝐴) we have 𝐴 = 𝑁0(𝑍) = 𝑁1(𝑍).
Building on Proposition5.45and using the same technique, de Brecht [deB13]
showed:
Proposition 5.46. Let 𝒳 and 𝒴 be topological spaces, with 𝒳 metrisable.
Let 𝜑 ∶ 𝒳 → 𝒴 be an open and continuous map with Polish fibres, 𝐴 ⊆ 𝒴
Since every second countable𝑇0space has an admissible representation which is open and has Polish fibres, we obtain:
Theorem 5.47 ([deB13, Theorem 78]). Let 𝒳be a second countable𝑇0 space, 𝜌 ∶⊆ 𝜔𝜔 → 𝒳 an admissible representation of 𝒳. For any countable 𝛼, 𝜉 > 0 and every 𝐴 ⊆ 𝒳 we have
𝐴 ∈ 𝐷𝜉(𝚺0𝛼(𝒳)) ⟷ 𝜌−1(𝐴) ∈ 𝐷𝜉(𝚺0𝛼(dom 𝜌)).
Proof. The left to right implication follows from the continuity of the admiss-ible representation and the fact that the preimage map 𝜌−1 is a complete Boolean homomorphism.
For the right to left implication, it is enough by Propositions 5.45 and 5.46 to show that we can assume 𝜌 to be open with Polish fibres – since such an admissible representation always exists by Theorem 5.17. So let 𝛿 ∶⊆ 𝜔𝜔 → 𝒳 be any admissible representation of 𝒳, then there exists a continuous 𝑓 ∶ dom 𝜌 → dom 𝜎with𝛿 ∘𝑓 = 𝜌on the domain of𝜌. If𝛿−1(𝐴) ∈ 𝐷𝜉(𝚺0𝛼(dom 𝛿)) then as in the first implication we have
𝜌−1(𝑆) = 𝑓−1(𝛿−1(𝑆)) ∈ 𝐷𝛼(𝚺0𝜃(dom 𝜌)).
This concludes the claim.
The proof of Proposition5.44 is now straightforward.
Proof of Proposition 5.44. Since 𝐷1(𝚺0𝛼)is just 𝚺0𝛼,(i) is a particular case of (ii). Let𝐵 ∈ 𝐷𝜉(𝚺0𝛼(𝒴))and suppose that𝐴 ⊆ 𝒳satisfies𝐴 ≼W 𝐵. Let𝜌𝒳, 𝜌𝒴 be admissible representations of𝒳, 𝒴respectively. Since𝐴 ≼W 𝐵, there exists a continuous𝑓 ∶ dom 𝜌𝒳→ dom 𝜌𝒴 with(𝜌𝒴∘ 𝑓)−1(𝐵) = 𝜌−1𝒳 (𝐴). By continuity, 𝜌−1𝒳 (𝐴) = (𝜌𝒴 ∘ 𝑓)−1(𝐵) ∈ 𝐷𝜉(𝚺0𝛼(dom 𝜌𝒳)), and so by Theorem 5.47 𝐴 is 𝐷𝜉(𝚺0𝛼) in𝒳.
In the Baire space, the pointclasses and the Wadge quasi-order are two sides of the same coin. Moreover while the pointclasses 𝚺0𝛼, 𝚷0𝛼 and 𝐷𝜉(𝚺0𝛼) are defined in terms of operations on open sets, Wadge showed that every non-self-dual Borel pointclass – i.e. a pointclass of Borel sets which is not closed under complementation – can be described in this fashion. Since we investigate a generalisation of the Wadge quasi-order on 𝜔𝜔 to more general spaces, it is therefore natural to wonder what is the generalisation of the notion pointclass.
With Proposition 5.44 in mind and by analogy with the case of 𝜔𝜔, our ap-proach suggests to define a ‘pointclass’ in any second countable𝑇0 space as an initial segment of the quasi-order ≼W.
It is worth mentioning that after the paper on which this chapter is based was finished we discovered that Louveau and Saint Raymond [LS11, Section 4]
define for each Borel non-self dual pointclass Γ in 𝜔𝜔 a corresponding class Γ(𝒳) for every metric separable space 𝒳 as follows, Γ(𝒳) is the family of those sets 𝐴 ⊆ 𝒳 such that for every (total) continuous map 𝑓 ∶ 𝜔𝜔 → 𝒳 we have 𝑓−1(𝐴) ∈ Γ. It is easy to see these classes Γ(𝒳) are always initial segments of our quasi-order ≼W. Conversely when 𝒳is Polish, one can show1
1Using that every Polish space has a total (i.e. defined on the whole𝜔𝜔) admissible rep-resentation as proved by Brattka [Bra99, Corollary 4.4.12].
that if Γ′ is a initial segment for ≼W consisting in Borel subsets of 𝒳 which is not closed under complementation, then there is a Borel non-self-dual point class Γ of 𝜔𝜔 such that Γ′ = Γ(𝒳). However we reserve the investigation of the relation between these classes and the quasi-order ≼W as well as a general discussion on the notion of pointclass in arbitrary spaces for a later work.