Admissible representations

A topological space 𝒳 is calledsecond countable if it admits a countable base of open sets. It satisfies the separation axiom 𝑇₀ if every two distinct points are topologically distinguishable, i.e. for any two distinct points𝑥 and𝑦 there is an open set which contains one of these points and not the other. It is called zero-dimensional, or 0-dimensional, if it admits a base of clopen sets, i.e. of simultaneously open and closed sets. A space is second countable and 0-dimensional if and only if it is homeomorphic to a subset of 𝜔^𝜔. A Polish space is a second countable completely metrisable topological space, the Baire space is a crucial example of Polish space. Recall [Kec95, (3.11), p.17] that a subspace of a Polish space is Polish if and only if it 𝚷⁰₂, i.e. a countable intersection of open sets.

Let 𝒳, 𝒴 be second countable 𝑇₀ spaces. If 𝐴 ⊆ 𝒳 and 𝑓 ∶ 𝐴 → 𝒴 is a function, 𝑓 is called a partial function from 𝒳 to 𝒴, in symbols 𝑓 ∶⊆ 𝒳 → 𝒴, and we refer to 𝐴 as the domain of 𝑓, denoted by dom 𝑓. A partial function 𝑓 ∶⊆ 𝒳 → 𝒴 is continuous if it is continuous on its domain for the subspace topology ondom 𝑓, i.e. if for every open𝑈 of 𝒴 there is an open 𝑉 of 𝒳such that 𝑓⁻¹(𝑈 ) = 𝑉 ∩ dom 𝑓.

We quasi-order the partial functions from 𝜔^𝜔 into 𝒳 by saying that for 𝑓, 𝑔 ∶⊆ 𝜔^𝜔 → 𝒳

𝑓 ⩽^rep 𝑔 ⟷ { there exists a continuous ℎ ∶ dom 𝑓 → dom 𝑔 with 𝑓(𝛼) = 𝑔 ∘ ℎ(𝛼) for all 𝛼 ∈ dom 𝑓.

Clearly, if 𝑔 is continuous and𝑓 ⩽^rep𝑔, then 𝑓 is continuous too. Hence the set of partial continuous functions from 𝜔^𝜔 into 𝒳 is downward closed with respect to⩽^rep.

Definition 5.14([Wei00]). Let𝒳be a topological space. A partial continuous function 𝜌 ∶⊆ 𝜔^𝜔 → 𝒳 is called an admissible representation of 𝒳 if it is a

⩽^rep-greatest element among partial continuous functions to 𝒳, i.e. 𝑓 ⩽^rep 𝜌 holds for every partial continuous𝑓 ∶⊆ 𝜔^𝜔 → 𝒳.

Observe that an admissible representation𝜌 of𝒳is necessarily onto𝒳, since for every point 𝑥 ∈ 𝒳, we have 𝑐_𝑥 ⩽^rep 𝜌 where 𝑐_𝑥 ∶ 𝜔^𝜔 → 𝒳, 𝛼 ↦ 𝑥 is the constant function.

Remark 5.15. Since the subspaces of𝜔^𝜔 are up to homeomorphism the second countable 0-dimensional spaces, an admissible representation of 𝒳 is also a continuous map 𝜌 ∶ 𝐷 → 𝒳 from some second countable 0-dimensional space 𝐷 such that for every continuous map 𝑔 ∶ 𝐸 → 𝒳 from a second countable 0-dimensional space 𝐸 there exists a continuous map ℎ ∶ 𝐸 → 𝐷 such that 𝑔 = 𝜌 ∘ ℎ.

Now it is well known that every second countable𝑇₀ space𝒳has an admiss-ible representation. Since it is simple and crucial for the sequel we now explain this fact in detail.

Definition 5.16. Let 𝒳 be a second countable 𝑇₀ space and (𝑉_𝑛)_𝑛∈𝜔 be a countable base of open sets for𝒳. We define thestandard representation of𝒳 with respect to(𝑉_𝑛) to be the partial map 𝜌 ∶⊆ 𝜔^𝜔 → 𝒳defined by

𝜌(𝛼) = 𝑥 ⟷ {𝑛 ∣ ∃𝑘 𝛼(𝑘) = 𝑛 + 1} = {𝑛 ∣ 𝑥 ∈ 𝑉_𝑛}.

Notice that in the definition 𝜌 is indeed a function on its domain because 𝒳 is 𝑇₀. An 𝛼 ∈ 𝜔^𝜔 codes via 𝜌 a point 𝑥 ∈ 𝒳 if and only if 𝛼 enumerates the indices of all the𝑉_𝑛’s to which 𝑥 belongs, while0 can be thought of as an index for the whole space 𝒳– which may not appear among the 𝑉_𝑛’s.

Theorem 5.17. For every second countable 𝑇₀ space 𝒳 there exists an ad-missible representation 𝜌 ∶⊆ 𝜔^𝜔 → 𝒳. Moreover it can be chosen such that

(i) 𝜌 is open,

(ii) for every 𝑥 ∈ 𝒳, the fibre 𝜌⁻¹(𝑥) is Polish.

Proof. Let (𝑉_𝑛) be a countable base for 𝒳 and let 𝜌 ∶⊆ 𝜔^𝜔 → 𝒳 be the standard representation of 𝒳 with respect to (𝑉_𝑛). It is enough to show that 𝜌 satisfies all the requirements.

Continuity: Note that 𝜌⁻¹(𝑉_𝑛) = {𝛼 ∈ dom 𝜌 ∣ ∃𝑘 𝛼(𝑘) = 𝑛 + 1} is open in 𝜔^𝜔 for every 𝑛, so 𝜌 is continuous.

Openness: For every basic𝑁_𝑠= {𝑥 ∈ 𝜔^𝜔∣ 𝑠 ⊆ 𝑥}, 𝑠 ∈ 𝜔^<𝜔, we have 𝜌(𝑁_𝑠) =

⋂𝑘<|𝑠|𝑉_𝑠

𝑘−1 (where𝑉₋₁ = 𝒳by convention) which is open in 𝒳, so 𝜌 is an open map.

Polish fibres: For every point𝑥 ∈ 𝒳

𝜌(𝛼) = 𝑥 ⟷ ∀𝑛[(∃𝑘 𝛼(𝑘) = 𝑛 + 1) ↔ 𝑥 ∈ 𝑉_𝑛] is a 𝚷⁰₂ definition of the fibre in 𝑥, so 𝜌 has Polish fibres.

Admissibility: Let𝑓 ∶⊆ 𝜔^𝜔→ 𝒳be continuous. Note that for every𝑛 ∈ 𝜔and every 𝛼 ∈ dom 𝑓

∃𝑘 𝑓(𝑁_𝛼↾

𝑘) ⊆ 𝑉_𝑛 ⟷ 𝑓(𝛼) ∈ 𝑉_𝑛.

Let𝑘 ↦ ((𝑘)₀, (𝑘)₁) be some bijection from 𝜔 to 𝜔 × 𝜔 whose inverse is denoted by⟨(𝑘)₀, (𝑘)₁⟩ = 𝑘. We defineℎ ∶ dom 𝑓 → dom 𝜌 by

ℎ(𝛼)(𝑘) = {(𝑘)₁+ 1 if 𝑓(𝑁_𝛼↾

(𝑘)0) ⊆ 𝑉_(𝑘)1,

0 otherwise.

Clearlyℎ is continuous. Moreover for every 𝛼 ∈ dom 𝑓 and every 𝑛 ∈ 𝜔, if 𝑓(𝛼) ∈ 𝑉_𝑛 then there exists 𝑘 ∈ 𝜔 such that 𝑓(𝑁_𝛼↾

𝑘) ⊆ 𝑉_𝑛 and so ℎ(𝛼)(⟨𝑘, 𝑛⟩) = 𝑛 + 1.

Conversely, if for𝛼 ∈ dom 𝑓 we haveℎ(𝛼)(𝑘) = 𝑛 + 1then it means that 𝑓(𝑁_𝛼↾

(𝑘)0) ⊆ 𝑉_𝑛 and so 𝑓(𝛼) ∈ 𝑉_𝑛.

It follows that𝑓(𝛼) = 𝜌 ∘ ℎ(𝛼) for all 𝛼 ∈ dom 𝑓, as desired.

Remark 5.18. i. If(𝑉_𝑛) is a countable base of a𝑇₀ space 𝒳, the map 𝜎 ∶⊆

𝜔^𝜔 → 𝒳defined by

𝜎(𝛼) = 𝑥 ⟷ Im 𝛼 = {𝑛 ∣ 𝑥 ∈ 𝑉_𝑛}.

is also an admissible representation of𝒳. Indeed 𝜎 is continuous and for 𝜌the standard representation of𝒳with respect to(𝑉_𝑛)we have𝜌 ⩽^rep 𝜎.

To see this we define a monotone map 𝜑 ∶ 𝜔^<𝜔 → 𝜔^<𝜔 by induction on the length by:

𝜑(∅) =∅

𝜑(𝑠^⌢0) = {𝜑(𝑠)^⌢ 𝜑(𝑠)(0) if 𝜑(𝑠) ≠ ∅

∅ otherwise,

𝜑(𝑠^⌢ (𝑛 + 1)) =𝜑(𝑠)^⌢ 𝑛.

This map𝜑induces a continuous map𝜑^∗ ∶ dom 𝜑^∗ → 𝜔^𝜔 where𝜑^∗(𝛼) =

⋃_𝑛∈𝜔𝜑(𝛼↾_𝑛)for𝛼 ∈ {𝛼 ∈ 𝜔^𝜔 ∣ lim_𝑛→∞𝜑(𝛼↾_𝑛) = ∞} = dom 𝜑^∗([Kec95,

(2.6), p.8]). We havedom 𝜌 ⊆ dom 𝜑^∗ for if 𝛼 ∈ dom 𝜌 then there exists 𝑛with𝜌(𝛼) ∈ 𝑉_𝑛, so there exists a𝑘 minimal such that𝛼(𝑘) > 0. Hence if 𝛼 = 0^{𝑘 ⌢} 𝛼(𝑘)^⌢ 𝛽 then 𝜑^∗(𝛼)(𝑛) = 𝛼(𝑛 + 𝑘) − 1 when 𝛼(𝑛 + 𝑘) > 0 and 𝜑^∗(𝛼)(𝑛) = 𝛼(𝑘) − 1 if 𝛼(𝑛 + 𝑘) = 0. Moreover we clearly have

= 𝜌 = 𝜎 ∘ 𝜑^∗.

ii. However, a standard representation 𝜌 of a second countable space 𝒳 relatively to some base has the following advantage. If one chooses the bijection𝑘 ↦ ((𝑘)₀, (𝑘)₁)from𝜔to𝜔 × 𝜔 such that(𝑘)₀ ⩽ 𝑘for every 𝑘, then the map ℎ ∶ dom 𝑓 → dom 𝜌 defined in the proof of Theorem 5.17 as a witness to the fact that 𝑓 ⩽^rep 𝜌 is actually Lipschitz, i.e. ℎ(𝛼)↾_𝑛 depends only on 𝛼↾_𝑛 for every 𝛼 ∈ dom 𝑓 and every 𝑛 ∈ 𝜔. This is sometimes convenient.

Importantly, Brattka [Bra99, Corollary 4.4.12] showed that every Polish space 𝑋 has a total admissible representation, i.e. an admissible representa-tion 𝜌 ∶⊆ 𝜔^𝜔 → 𝑋 with dom 𝜌 = 𝜔^𝜔. As an easy consequence one gets that for every second countable 𝑇₀ space 𝑋: there exists an admissible representa-tion of 𝑋 with a Polish domain if and only if there exists a total admissible representation of 𝑋. Motivated by the rich theory of Polish spaces, it is nat-ural to consider the class of those second countable 𝑇₀ spaces which have a total admissible representation. As a matter of fact de Brecht [deB13] showed that this class coincides with the class of quasi-Polish spaces that he recently introduced. Moreover he showed that many classical results of descriptive set theory can be generalised to this large class of non necessarily Hausdorff spaces and that the metrisable quasi-Polish spaces are exactly the Polish spaces.

The real lineℝ is certainly the most important example of non zero-dimen-sional Polish space and we now introduce two different admissible representa-tions for it.

Example 5.19. Let (𝑞_𝑛)_𝑛∈𝜔 be an enumeration of the rationals and let 𝐼_𝑛 = (𝑞_𝑛

0, 𝑞_𝑛

1)be an enumeration of the non empty intervals of the real lineℝ with rational endpoints.

We define 𝜌_ℝ ∶⊆ 𝜔^𝜔 → ℝ relatively to the enumerated base(𝐼_𝑛)_𝑛∈𝜔 by 𝜌_ℝ(𝛼) = 𝑥 ⟷ Im 𝛼 = {𝑛 ∣ 𝑥 ∈ 𝐼_𝑛},

so that 𝛼 ∈ 𝜔^𝜔 codes 𝑥 ∈ ℝ if and only if 𝛼 enumerates all the intervals with rational endpoints in which 𝑥 belongs.

The second admissible representation is based on Cauchy sequences and it works mutatis mutandis for every separable complete metric space.

Example 5.20. Let (𝑞_𝑛)_𝑛∈𝜔 be an enumeration of the rationals, and let 𝑑 be the euclidean metric on ℝ. A sequence (𝑥_𝑘)_𝑘∈𝜔 is said to be rapidly Cauchy if for every 𝑖, 𝑗 ∈ 𝜔, 𝑖 < 𝑗 implies 𝑑(𝑥_𝑖, 𝑥_𝑗) ⩽ 2^−𝑖. The Cauchy representation 𝜎_ℝ ∶⊆ 𝜔^𝜔 → ℝ of the real line is defined by

𝜎_ℝ(𝛼) = 𝑥 ⟷ (𝑞_𝛼(𝑘))

𝑘∈𝜔 is rapidly Cauchy and lim

𝑘→∞𝑞_𝛼(𝑘)= 𝑥.

This is an admissible representation of ℝ.

To illustrate our ideas in the non-metrisable case we consider the Scott Do-main 𝒫𝜔, namely the powerset of 𝜔 partially ordered by inclusion and en-dowed with the Scott topology. A base of 𝒫𝜔 is given by sets of the form 𝑂_𝐹 = {𝑋 ⊆ 𝜔 ∣ 𝐹 ⊆ 𝑋} for some finite 𝐹 ⊆ 𝜔. This space is universal among the second countable𝑇₀ spaces. Indeed for every𝑇₀ space𝒳with some count-able base (𝑉_𝑛)_𝑛∈𝜔 the map 𝑒 ∶ 𝒳 → 𝒫𝜔, 𝑥 ↦ {𝑛 ∣ 𝑥 ∈ 𝑉_𝑛} is an embedding.

Example 5.21. The enumeration representation of 𝒫𝜔 is the total function 𝜌_En ∶ 𝜔^𝜔→ 𝒫𝜔 defined by

𝜌_En(𝑥) = {𝑛 ∣ ∃𝑘 𝑥_𝑘= 𝑛 + 1}.

It is easy to see that 𝜌_En is an open admissible representation with Polish fibres.

As another example of an admissible representation of 𝒫𝜔 consider:

Example 5.22. Let (𝑠_𝑛)_𝑛∈𝜔 be an enumeration of the finite subsets of 𝜔. We define 𝜌_<∞ ∶⊆ 𝜔^𝜔 → 𝒫𝜔by

𝜌_<∞(𝛼) = 𝑥 ⟷ ∀𝑛 ∈ 𝜔 𝑠_𝛼(𝑛)⊆ 𝑠_𝛼(𝑛+1) and ⋃

𝑛∈𝜔

𝑠_𝛼(𝑛) = 𝑥.

The domain of𝜌_<∞ is closed and 𝜌_<∞ is clearly continuous. The map𝜌_<∞ is also an admissible representation of the space 𝒫𝜔 since it is continuous and 𝜌_En ⩽^rep 𝜌_<∞, as witnessed by the continuous 𝑓 ∶ 𝜔^𝜔 → dom 𝜌_<∞ defined by

𝑓(𝛼)(𝑛) = 𝑘, where 𝑠_𝑘 = {𝑚 ∣ ∃𝑗 ⩽ 𝑛 𝛼(𝑗) = 𝑚 + 1}.