Parovičenko spaces with structures

We study an analogue of the Parovičenko property in categories of compact spaces with additional structures. In particular, we present an internal characterization of this property in the class of compact median spaces.


Introduction
A compact space K is said to be Parovičenko if for every continuous surjections f : A → B and g : K → B, where A, B are compact metrizable spaces, there exists a continuous surjection h : K → A satisfying f • h = g. This definition naturally generalizes to higher weights, leading to the notion of a κ-Parovičenko space, where κ is a cardinal bounding the weights of the spaces A, B. It is also natural to consider the same property in other categories. We shall concentrate on categories of structured compact spaces, defined below. We shall present a general (and rather standard) result on the existence of structured Parovičenko spaces. Next, we discuss one particular example: compact median spaces. The main result is an internal characterization of Parovičenko objects in this category.
The concept of a Parovičenko space was introduced by van Douwen and van Mill [14], where the authors actually used its internal properties, instead of the external categorytheoretic condition stated above. Originally, Parovičenko [10] showed that every compact space of weight ℵ 1 is a continuous image of theČech-Stone remainder N * of the natural numbers; he also proved that, assuming the Continuum Hypothesis, this is the unique Parovičenko compact space of weight c.
Later on, Negrepontis [9] generalized Parovičenko's results to higher cardinals; some of his questions were several years later answered by Dow [2]. Finally, van Douwen and van Mill [13] showed that the Continuum Hypothesis is equivalent to the existence of a unique Parovičenko space of weight ℵ 1 . At the same time, Błaszczyk and Szymański [1] gave a simpler proof of Parovičenko theorem, also finding another characterization of N * , under the Continuum Hypothesis. Some further simplifications were subsequently found by Engelking [4]. Nowadays, Parovičenko spaces (dually, Parovičenko Boolean algebras) belong to the folklore of topology and set theory.
The aim of this note is to extend this concept to categories of compact spaces with structures, emphasizing one particular case: compact median spaces. This class was studied, in the context of Parovičenko theorem, by van Mill [15], who showed that none of the obvious subspaces of the superextension of the natural numbers (that plays the role of theČech-Stone compactification of N in the category of compact median spaces) has the Parovičenko property. Our main results are: the existence of Parovičenko median spaces and an internal characterization of such spaces. We finally show that the Continuum Hypothesis is equivalent to the uniqueness of a Parovičenko median space of weight 2 ℵ 0 , an analogue of the result of van Douwen and van Mill [14].
All topological spaces under consideration are assumed to be at least Hausdorff. We shall use standard terminology concerning general topology and category theory.

Structured compact spaces
In this section we present category-theoretic approach to compact spaces with additional structures. We first explain this concept informally, the precise definition is given below. Roughly speaking, compact spaces with structures are objects of the form K , a , where a is a structure on K , so far with no particular meaning. Given two compact spaces with structures K , a and L , b , we would like to consider continuous mappings f : K → L that preserve the structures. If the structures are not clearly defined, the only thing we can say is that structure-preserving maps are some particular continuous maps. It is natural to require that identities should be structure-preserving and compositions of structure-preserving maps should be structure-preserving. Here we arrive at the proper concept: Structured compact spaces can be defined by a faithful functor from some (possibly very abstract) category K into the category of compact spaces Comp.
To be more precise, by Comp we shall understand the category whose objects are nonempty compact spaces and arrows are continuous surjections (i.e. quotient maps). Restricting attention to surjective maps is convenient for our applications, especially when dealing with inverse sequences, although there is no problem to consider arbitrary continuous maps.
Recall that a functor F : K → L is faithful if it is one-to-one on hom-sets, namely, given morphisms f : a → b, g : a → b, where a, b are objects of K, the equality F( f ) = F(g) implies f = g.
Fix a faithful covariant functor : K → Comp. We can look at the graph of as a category whose objects are pairs of the form K , a , where a is a K-object such that K = (a) and an arrow from K , a to L , b is a K-arrow f : a → b, identified by with a continuous surjection ( f ) : K → L. The fact that is faithful simply means that f is uniquely determined by ( f ). In other words, we can call a and b structures on K and L, respectively, and say that ( f ) is a structure-preserving continuous map. We shall talk about -structured compact spaces, having in mind objects of the form K , a , where a is a K-object such that K = (a). For simplicity, we shall say that g : Summarizing, a category of structured compact spaces is given just by a faithful functor into the category of compact spaces. This clearly covers all algebraic structures as well as many more abstract ones.
For example, consider the category K of all Banach spaces with linear isometric embeddings. Given a Banach space X , let (X ) be the dual unit ball of X endowed with the weak-star topology. It is well-known that this is a faithful contravariant 1 functor into Comp. Thus, dual balls of Banach spaces can be viewed as compact spaces with structures, where the structure is actually a Banach space, even though all the information is "contained" in its dual unit ball.
All the considerations above belong to the topic of concrete categories, the only formal difference is that we deal with functors into the category of compact spaces instead of the category of sets.

Parovičenko spaces
From now on we fix a faithful covariant functor : K → Comp. We shall work in the category of -structured compact spaces. It is natural to define the weight of a -structured compact space K , a to be just the weight of K .
From now on, we fix a regular cardinal κ ℵ 0 . We shall say that a -structured compact space P, π is κ-Parovičenko if for every -preserving surjection f : L , b → K , a of spaces of weight < κ, for every -preserving surjective map p : P, π → K , a there exists a -preserving surjection q : P, π → L , b such that p = f • q.
In order to show the existence and good properties of Parovičenko spaces it is necessary to make some natural assumptions on the functor .
Namely, from now on we shall assume that satisfies the following conditions: There exist, up to isomorphism, at most κ <κ many -structured compact spaces of weight < κ. (C) Given an infinite cardinal λ, every inverse sequence of -preserving surjections between spaces of weight < λ has a limit in the category of -preserving surjections; if the length of this sequence is < cf(λ) then the weight of the limit is < λ. (D) Given an uncountable cardinal λ, every -structured compact space of weight λ κ is the limit of a continuous inverse sequence of length λ consisting of -structured compact spaces of weight < λ and with -preserving surjections. (F) Given a regular cardinal λ > ℵ 0 , given an inverse sequence X = X α , p β α , λ ofpreserving surjective maps between spaces of weight < λ, given a -preserving surjec-tion f : lim X → Y , there exists α < λ and a -preserving surjection f : X α → Y such that f = f • p α , where p α is the canonical projection onto X α . (T) There exists a -structured compact space E, e such that every -structured compact space has a -preserving surjection onto E.
It should be clear that the identity functor of Comp satisfies conditions (A)-(T). Other natural functors, coming from algebraic structures on compact spaces, are described in Sect. 2.2 below. In conditions (C), (D) and (F) λ is an arbitrary infinite cardinal. Condition (A) will be sometimes called the reversed amalgamation property. Condition (F) will be called the factorization property. Condition (T) (the existence of a weakly terminal object) is assumed for the sake of convenience only, it could be replaced by a weaker one saying that for every -structured compact spaces X , Y there exists a -structured compact Z and -preserving surjective maps f : The next fact belongs to the folklore, we sketch its proof only for completeness.

Lemma 2.1
Let K be a compact space of weight κ <κ and let L be a compact space of weight < κ. Then the set of all surjective maps from K onto L has cardinality κ <κ .
Proof Let Z (K ) denote the lattice of all closed G δ subsets of K . Then |Z (K )| (κ <κ ) ℵ 0 = κ <κ . Fix a closed basis B in L of cardinality μ < κ that consists of closed G δ sets. Every surjective map f : K → L is uniquely determined by its inverse restricted to B, which is an embedding of B into the lattice Z (K ). Finally, Z (K ) has at most (κ <κ ) μ = κ <κ subsets of cardinality μ. We define inductively an inverse sequence S = K α , p β α ; α < κ <κ of -structured compact spaces whose bonding maps are -preserving surjections. We start with K 0 = K and at each limit stage we use condition (C). Given α < κ <κ , we require that there exists a -preserving surjection r α : K α+1 N α such that the following diagram commutes. This is possible, because of the reversed amalgamation (condition (A)) for the mappings q α • p α 0 , f α . Finally, we let L to be the limit of this sequence in the category of -structured compact spaces, which exists by condition (C). The same condition applied to λ = (κ <κ ) + says that w(L) κ <κ . Finally, by the construction it is clear that condition (L) holds.
Proof We construct a -continuous inverse sequence of length κ + , starting from a weakly terminal object given by condition (T). At each successor step we use Lemma 2.2, while at limit steps we use condition (C). Finally, the factorization property of (condition (F) applied to λ = κ + ) shows that the limit of this sequence is κ-Parovičenko.
The following statement is an adaptation of well-known results in Fraïssé-Jónsson theory, see e.g. [3] or [7].

Theorem 2.4 Let P be a κ-Parovičenko
(3) If κ is regular then there exists at most one (up to isomorphism) κ-Parovičenkostructured compact space of weight κ.
Proof (1): First, suppose that w(K ) < κ. By (T), there are -preserving surjections f : P E and g : K E, where E is a weakly terminal -object specified in condition (T). By the definition of a κ-Parovičenko space, we get a -preserving surjection q : where the sequence is continuous, all bonding maps are -preserving surjections and w(K α ) < κ for each α < κ. We may assume that K 0 = E. Suppose we have constructed -preserving surjections p α : P K α for α < δ, where δ < κ is a fixed ordinal, such that q β α • p β = p α for every α < β < δ. If δ is a limit ordinal, we use the continuity of the sequence to construct p δ . If δ = β + 1 then we find p δ just using the definition of a κ-Parovičenko -structured compact space. Finally, the surjective map p : P K is the limit of the sequence { p α } α<κ . This completes the proof of (1).
(2) and (3): We shall prove both statements simultaneously. Namely, assume P and Q are two -structured κ-Parovičenko compact spaces of weight κ.
where both sequences are continuous, all bonding maps arepreserving surjections and all spaces X α , Y α have weight < κ. Now observe that the inverse sequence X α , p β α , κ has the following property: (P) Given α < κ, given a -preserving surjection f : K X α with w(K ) < κ, there exist β > α and a -preserving surjection g : X β K such that This is easily proved by applying the κ-Parovičenko property and the factorization property (F). Obviously, the sequence Y α , q β α , κ satisfies the same condition. Now suppose for the moment that X 0 = Y 0 = K . A standard back-and-forth argument gives the required -isomorphism from P to Q. This is visualized in the following commutative diagram: The successor steps are made by applying condition (P) and the limit stages are taken care by condition (C). Now, assume that K = E (a weakly terminal object from condition (T)). Then the backand-forth argument above shows that P is isomorphic to Q, proving (3).
Finally, in order to show (2), let us assume that P = Q, X α = Y α and p β α = q β α in the diagram above. Using the factorization property (F), we may assume that there are α 1 , β 1 The same back-and-forth argument sketched in the diagram above gives the required isomorphism h satisfying f = g • h.

Topological algebras
We now make a brief discussion of possible applications of the concepts of this section. Namely, fix a countable first-order language L consisting of function symbols (i.e. algebraic operations) and fix a class K of compact L-algebras, that is, if K ∈ K then K is a compact Hausdorff space which is at the same time an L-model such that all L-operations in K are continuous. Let us assume that K is closed under standard products, that is, a product of a family F ⊆ K is the usual product F with the Tikhonov topology, and all L-operations are defined coordinate-wise. Let us also assume that K is stable under closed subalgebras. In other words, if K ∈ K and K ⊆ K is a closed subspace that is also closed under all L-operations, then K ∈ K.
These two assumptions ensure that K has inverse limits, that are actually inverse limits in the category of compact spaces.
In order to describe everything in the language of -structured compact spaces, we consider the natural forgetful functor from K into Comp, where K is viewed as the category of all continuous surjections that preserve all L-operations.
It follows that K (or formally the just-defined functor ) satisfies conditions (A), (B) and (C). In fact, (C) follows from the remarks above; (B) is a standard "counting" argument (using the fact that the language is countable); (A) can be deduced from the fact that K has products and is closed under substructures. More precisely, given continuous epimorphisms of L-structures f : X → Z , g : Y → Z , their pullback belongs to the category K. Recall that the pullback of f, g is a pair of maps f , g , where f : W → X , g : W → Y are defined as follows: Condition (T) is satisfied as long as there exists an L-algebra E in K such that every other K ∈ K has a continuous homomorphism onto E. Finally, condition (D) requires an addi-tional assumption. Namely, we need to know that K is closed under continuous homomorphic images. More precisely, given K ∈ K, given a compact L-algebra L, if there exists a continuous epimorphism f : K → L, then L ∈ K. Summarizing, we have: Theorem 2.5 Let L be a countable first-order language consisting of function symbols only and let K be a class of compact L-algebras with the following properties: (1) K is stable under products and closed subalgebras, Proof In view of the remarks above, only condition (D) requires a proof. This is a rather standard closing-off argument, although we were unable to find it explicitly written in the literature, therefore we sketch it below. We use the method of elementary submodels and we refer to [5, Chapter 17] for more details and explanations. Namely, given a compact space K and given an elementary submodel M of a big enough H (θ ) (the family of sets of hereditary cardinality < θ), there is a natural equivalence relation The quotient space with respect to this relation is denoted by K /M and the quotient map is denoted by q M : K → K /M. The crucial fact is that K /M is an L-algebra and q M is a homomorphism, whenever M "knows" the L-operations of M. This is satisfied whenever L ∈ M is countable.
Note that the weight of K /M is |M| and the Skolem-Löwenheim theorem says that there exists a countable elementary submodel M of H (θ ) such that K and any fixed countable set is in M.
Finally, we need to know that if θ is big enough so that K ∈ H (θ ), then the family of all elementary substructures of H (θ ) that contain K and all the L-operations on K is directed and closed under unions of arbitrary chains. A suitable chain of elementary submodels of H (θ ), starting with a countable one, provides an inverse sequence witnessing (D).
One should admit that condition (T) is usually trivial to check, therefore it has been ignored in the statement above.

Compact median spaces
A topological space is supercompact if it has a subbase B for its closed sets (called a binary subbase) such that every linked family F ⊆ B has nonempty intersection. A family F is linked if A∩ B = ∅ for every A, B ∈ F . By Alexander's subbase lemma, every supercompact space is compact. A nontrivial result of Strok and Szymański [11] says that every compact metric space is supercompact. By the result of van Douwen and van Mill [13], every infinite supercompact space contains non-trivial convergent sequences, therefore not all compact spaces are supercompact. We say that two sets A, A topological space X which possesses a normal binary subbase is called normally supercompact. For more information about supercompactness we refer to van Mill's book [15]. We now discuss briefly convexity structures in median spaces. For more details we refer to van de Vel's monograph [12].
Supercompact spaces, especially normally supercompact spaces, have some nice geometric properties, in the sense of general convexity structures. Recall that an interval convexity (convexity for short) on the set X is the family G of subsets of X which satisfies the following conditions: Elements of G are called convex sets. A convex set whose complement is convex is called a halfspace. By (2) we can define the convex hull conv A of any set A ⊆ X as the set {F ∈ G : A ⊆ F}. We write [a, b] instead of conv{a, b} and call it the segment joining a and b.
Let X be a normally supercompact space with a fixed normal binary subbase B. The interval map I B : X × X → P(X ) is defined by the formula: It is not hard to see that the family Since each element of B is convex, we see that every normally supercompact space satisfies the condition: CC 2 : each two distinct points can be screened with closed convex sets.
Let us note that the convexity G I B is binary in the sense that each finite linked subfamily of G I B has nonempty intersection (see [15,Theorem 1.3.3]). So, each normally supercompact space is a compact space with a binary convexity satisfying CC 2 . It turns out that the converse is true as well. If X is a compact space with a binary convexity G satisfying CC 2 then, by compactness, the collection of all closed convex sets is a closed subbase for the topology of X . Moreover, the collection of all closed convex sets is normal (see [6, Proposition V.1.2]). In conclusion, the following two classes topological spaces are equal: • compact spaces with a binary convexity satisfying CC 2 • normally supercompact spaces. Such spaces will be called compact median spaces.

Basic properties of compact median spaces
In the class of compact median spaces the following counterpart of Urysohn lemma is true (see [16,  The map f from the last theorem is a special case of a more general concept of convexity preserving map. Namely f : X → Y , is called a convexity preserving map, if f −1 (G) ∈ F for each G ∈ G, where X and Y are spaces with convexities F and G, respectively. In case of binary convexities, this is equivalent to (cp) For median spaces, convexity preserving maps are precisely the median preserving maps (with the obvious meaning). Indeed, a convexity preserving map f necessarily preserves the median, because of (cp). On the other hand, if f is median preserving and , showing that f satisfies (cp). From now on, we will consider the category CM consisting of all compact median spaces and surjective continuous median preserving maps. Such maps will be called epimorphisms and denoted f : X Y for X, Y ∈ CM. Recall that a subset X of a median space M is median-closed (or median-stable) if m(x, y, z) ∈ X whenever x, y, z ∈ X . It follows from Proposition 3.1 that:

Theorem 3.3 Each compact median space X of weight κ
ℵ 0 can be represented as the limit of an inverse sequence of compact median spaces X α ; p β α ; α ≤ β < κ , where:

Each bonding map p
β α is convexity preserving. 5. If X is zero-dimensional then so is each X α and moreover for each α < κ there exist closed convex sets A α , and p α+1 α is the projection.

Lemma 3.4 Let K , L be compact median spaces, where K is zero-dimensional of weight
κ ℵ 0 , and let f : K → L be an epimorphism. There exists an inverse sequence of compact median spaces S = Y α , f β α , α < β < κ such that K = lim ← − S and is an inverse sequence of compact median zero-dimensional spaces such as in the Theorem 3.3.
. Let Y 0 = L and h 0 = f . If α < κ is a limit ordinal then let h α be the map induced by {h β : β < α} and let Since all h α are epimorphisms, the induced map h : K → Y is an epimorphism. We claim that h is one-to-one, i.e. it is an isomorphism. Indeed, if x, y ∈ K are different points, then there is α < κ and clopen sets Proof Let R be a normal subbase both for the closed sets and for the convexity and let B ⊆ R be a subbase for the closed sets. The convexity C generated by the family B satisfies the condition CC 2 . We check that C ⊆ C . Let C ∈ C. Define I : X 2 → X and I : Let x ∈ X and let C ⊆ X \{x} be a closed convex set. By CC 2 , the intersection {[x, c] : c ∈ C} ∩ C is a singleton, say {c 0 }. Again by CC 2 , there are A, B ∈ B such that x ∈ B and c 0 ∈ A and A ∪ B = X . Finally, C ∩ A = ∅, because if c ∈ C ∩ A then c 0 ∈ [x, c] ⊆ A, which would be a contradiction.

Parovičenko median spaces
Let κ be a fixed infinite regular cardinal. We shall now discuss κ-Parovičenko median spaces, aiming at their internal characterization.  , without any extra settheoretic assumptions there exists a unique ω-Parovičenko compact median space P ω of countable weight. It is an easy exercise to check that P ω carries the topology of the Cantor set. On the other hand, its median structure seems to be rather complicated and not definable by any simple formula.

Main result
Let P be a compact median space. The collection of all clopen halfspaces in P will be denoted by H (P) and let H (P) + = H (P)\{∅}. Recall that H (P) + separates the points of P. Given a family A ⊆ H (P), it is natural to consider the following quotient space P/A: Given x, y ∈ P, we define x ∼ y iff no member of A separates x from y. The space P/A is formally the quotient P/ ∼, endowed with the quotient topology. It is actually a median space, where the median is well-defined by the formula m( x, y, z ∈ P. The canonical quotient map is obviously median preserving. We shall use this construction in the proof below. Note that the topological quotient space P/A can be defined as long as A is a family of clopen subsets of P. The fact that A consists of halfspaces allows us to get a median structure on the quotient. Note also that P/H (P) = P.
(c) ⇒ (b) Let f : P → K be an epimorphism with w(K ) < κ and let E, F ⊆ K be closed convex sets with E ∪ F = K . Since w(K ) = τ < κ, there exists a dense subset D ⊆ E ∩ F such that |D| ≤ τ.

Claim 4.5
Given an open set W ⊆ K whose complement is convex, there exists a family Proof of Claim 4.5 By [6, Corollary V.1.3], the collection R of closed halfspaces is a normal subbase both for the closed sets and for the convexity. Let B ⊆ R be a normal subbase for the closed sets such that |B| = τ . By Lemma 3.6, B generates the convexity structure of K . Moreover, any x ∈ X and closed convex set C ⊆ X \{x} can be screened with some A, B ∈ B. Given x ∈ W , let A, B ∈ B be such that x ∈ A ⊆ W and A ∪ B = K . Hence, there exists a collection of closed halfspaces {U α : α ∈ τ } such that W = {U α : α ∈ τ }. By Lemma 3.5 applied to the closed convex disjoint sets Proof of Claim 4.6 Let p ∈ P. By Claim 4.5, there is a collection of clopen halfspaces is a normal binary family, the canonical quotient mapping q : P → L is an epimorphism (i.e. it is median preserving). Define E = {L\A + : A ∈ A} and F = {L\B + : B ∈ B}. The sets E, F are closed convex and E ∪ F = L. Put K = E ×{0}∪ F ×{1} and define an epimorphism f : K → L by f (x, i) = x. By condition (a), there is an epimorphism g : P → K such that q = f • g. Now we shall show that Next we show (M2).
Let A ⊆ H (P) be a linked family and |A| < κ. Put L = P/(A ∪ A ), where A = {P\A : A ∈ A} and A ∪ A is a normal binary family. Let q : P L be the canonical epimorphism. Since A is linked and H (P) is binary we have {A + : A ∈ A} = ∅. Let Then f is an epimorphism. By condition (a) there is an epimorphism g : P → K such that q = f • g. We shall prove that g −1 ({x} × {1}) ⊆ A. Let a ∈ g −1 ({x} × {1}) and suppose that a ∈ A for some A ∈ A. Then a ∈ P\A and hence q(a) ∈ (P\A) + . This is a contradiction with q(a) = f (g(a)) = f (x, 1) = x ∈ {A + : A ∈ A}.
The same proof as in Lemma 2.2 works.
Proof We define an inverse sequence K α ; p β α ; α ≤ β < κ , where the spaces K α are compact median of weight ≤ 2 τ and the maps p β α are epimorphisms, in following way. Let K 0 = D 2 τ . Assume that α < κ and spaces K β for β < α and p β γ for γ ≤ β < α are already defined. If α is a limit ordinal let us define K α = lim ← − K β , p β γ ; γ ≤ β < α and p α β as a projection from K α onto K β for β < α. If α = β + 1 and α is an odd ordinal then we define K α and p α β as in Lemma 2.2. If α = β + 1 and α is an even ordinal then we define K α = K β × D 2 τ and p α β is a projection on second coordinate. Let P = lim ← − K α ; p β α ; α ≤ β < 2 τ . The space P is compact median and each projection p α : P → K α is an epimorphism. Obviously w(P) = 2 τ . Let q : P L and f : K L be epimorphisms, where w(K ) ≤ τ . Since w(L) ≤ τ and κ is regular, there is an even ordinal α < κ and an epimorphism q : K α L such that q = q • p α . By the construction, there is an epimorphism p : K α+1 K such that q • p α+1 α = f • p. It remains to note that which completes the proof that P is a τ + -Parovičenko space.