Vietoris hyperspaces of scattered Priestley spaces
aa r X i v : . [ m a t h . GN ] J u l Dedicated to the memory of Matatyahu Rubin (1946-2017)
VIETORIS HYPERSPACES OVER SCATTERED PRIESTLEY SPACES
TARAS BANAKH, ROBERT BONNET, WIESŁAW KUBIŚ (JULY 15TH, 2020)
Abstract.
We study Vietoris hyperspaces of closed and final sets of Priestley spaces. Weare particularly interested in Skula topologies. A topological space is
Skula if its topology isgenerated by differences of open sets of another topology. A compact Skula space is scatteredand moreover has a natural well-founded ordering compatible with the topology, namely, it isa Priestley space. One of our main objectives is investigating Vietoris hyperspaces of generalPriestley spaces, addressing the question when their topologies are Skula and computing theassociated ordinal ranks. We apply our results to scattered compact spaces based on certainalmost disjoint families, in particular, Lusin families and ladder systems.
Contents
1. Introduction 2Preliminaries 22. Priestley spaces and Vietoris hyperspaces 93. Interplay between (Cantor-Bendixson) height and (well-founded) rank 134. Canonical and tree-like Skula spaces and Applications to some classes 174.1. The space of initial subsets of a partial ordering 244.2. Mrówka spaces 264.3. Lusin families and ladder systems 275. Complements on Hyperspaces and on Skula spaces 295.1. Priestley hyperspaces versus Vietoris hyperspaces 295.2. Skula spaces and compact semilattices 316. Final remarks and open questions 33References 34
Date : (July 15th, 2020).1991
Mathematics Subject Classification. primary ), 54G12, 54D30, 06E05, ( secondary ). Key words and phrases.
Well-generated Boolean algebras, Priestley space, Vietoris hyperspace, Mrówkaspace, Lusin family, ladder system, Well ordering.The first author has been partially supported by NCN grant DEC-2012/07/D/ST1/02087.Research of the second author was supported by the Institute of Mathematics of the Czech Academy ofSciences. Introduction
This paper is motivated by a problem of detecting scattered compact spaces that are home-omorphic to (or embed into) scattered compact topological semilattices. This problem hasbeen addressed by T. Banakh, O. Gutik and M. Rajagopalan in [25] and [8]. The papers ofR. Bonnet and M. Rubin [13], and of A. Dow and S. Watson [21], develop classes of compactscattered spaces with a closed partial ordering. Such orders are “well-founded”. We study therelationship between the Cantor-Bendixson height and the well-founded rank in these spaces.A topological space is called a
Skula space if its topology is generated by differences of opensets of another topology on the same set. This concept was invented by Ladislav Skula [43,Result 2.2] in 1969, answering a purely category-theoretic question. Much later, in 1990,Dow and Watson [21] observed that compact Skula spaces are scattered and they related thisconcept to well-generated Boolean algebras introduced by Bonnet and Rubin [13].A
Priestley space is a compact 0-dimensional space endowed with a closed partial order suchthat clopen final subsets separate points (a set S is final if x ∈ S , x < y imply y ∈ S ). Priestleyduality [40] establishes the correspondence between Priestley spaces and bounded distributivelattices, exactly in the same manner as Stone duality between compact 0-dimensional spacesand Boolean algebras. In fact, every compact 0-dimensional space is Priestley when endowedwith the trivial ordering, therefore Priestley duality “contains” Stone duality.As it happens, a compact Skula space has a natural closed partial ordering that is well-founded and makes it a Priestley space. Our aim is to give more insight into this relation.In particular, we look at Vietoris hyperspaces of Priestley spaces, investigating when theirtopology is Skula. In that case, scatteredness and well-foundedness are intimately linked,and we study for a Skula space space and its Vietoris hyperspace the relationship betweenits Cantor-Bendixson height and its well-founded rank. We also consider a specialization:canonically Skula spaces. As we have mentioned above, one of our motivations is topologicalsemilattice theory in which Priestley spaces play a significant role: The free semilattice over aPriestley space is its Vietoris hyperspace (see below for more details). Preliminaries. A (join) semilattice , is a set X endowed with a binary operation, ∨ : X × X → X : the operation ∨ should be associative, commutative, and idempotent (in the sense that x ∨ x = x for all x ∈ X ). Every semilattice has a natural partial ordering: x ≤ y if and onlyif x ∨ y = y . By a topological semilattice we mean a Hausdorff topological space X endowedwith a continuous join semilattice operation ∨ . In that case ≤ is closed (as subset of X × X ).A poset is a partially ordered set. For a point p of a poset P the set ↓ p = { q ∈ P : q ≤ p } IETORIS HYPERSPACES OVER PRIESTLEY SPACES . . . (July 15th, 2020) 3 is called the principal ideal generated by p . We say that F ⊆ P is an initial subset wheneverfor every p ∈ P : if p ∈ F then ↓ p ⊆ F . For instance, if A ⊆ P then ↓ A := S p ∈ A ↓ p is aninitial subset of P .Also ↑ p := { q ∈ P : q ≥ p } for p ∈ P is called the principal filter generated by p , and ↑ A := S p ∈ A ↑ p is called a final subset of P .A poset P is called a linear ordering or a chain if the elements of P are pairwise comparable.A subset A of P is called an antichain if A consists of pairwise incomparable elements.A poset h P, ≤i is well-founded if each nonempty subset of P has a minimal element, that is P has no strictly decreasing sequence. A well-ordering is a well-founded linear ordering.Following [21], we call a Hausdorff topological space X a Skula space if the topology of X is generated by the base { U \ V : U, V ∈ T } for some topology T on X . A typical example ofa (non-scattered) Skula space is the Sorgenfrey line whose topology is generated by the base { U \ V : U, V ∈ T } for the topology T = {∅ , R } ∪ { ( −∞ , a ) : a ∈ R } .The following characterization of Skula spaces combines some known results of Bonnet andRubin [13] and of Dow and Watson [21] with some new results proved in this paper. To statethe theorem, following [13, §2.3], we introduce the following notion. For a space X , we saythat a family U := { U x : x ∈ X } is a clopen selector or more simple selector for X if each U x is a clopen (i.e. closed and open) subset of X and if U satisfies:(1) x ∈ U x for every x ∈ X ,(2) for any distinct x, y ∈ X either x / ∈ U y or y / ∈ U x , and(3) for any x ∈ X and y ∈ U x we get U y ⊆ U x .Given that every U x contains x , conditions (2) and (3) are equivalent to:(4) the relation x < U y ⇐⇒ x = y and x ∈ U y is irreflexive and transitive.Therefore a clopen selector U := { U x : x ∈ X } for X induces a partial order relation ≤ U on X , defined by x ≤ U y if and only U x ⊆ U y . If U is understood from the context, then ≤ U is denoted by ≤ and thus U x = ↓ x is the clopen principal ideal for any x ∈ X : in particular every principal ideal is clopen.Skula spaces were introduced (independently of Skula) by Bonnet and Rubin [13] in thealgebraic way as “well-generated Boolean algebras”. For a compact space X , we denote by Clop( X ) the set of clopen subsets of X . TARAS BANAKH, ROBERT BONNET, WIESŁAW KUBIŚ (JULY 15TH, 2020) A well-generated Boolean algebra is a Boolean algebra that has a sublattice (that is, a subsetclosed under meet and join) that generates the algebra and that has no strictly decreasingsequence for ≤ .By Theorem 2.1(i) ⇔ (iv), X is a compact Skula space if and only if Clop( X ) is well-generated.Equivalently, B is a well-generated Boolean algebra if and only if its space Ult( B ) of ultrafiltersis a Skula space. This equivalence was proved in a preprint of Bonnet and Rubin publishedin [13, Proposition 2.15(b)] and independently by Dow and Watson [21, Theorem 1]. ButLemma 4 and Theorem 3 in [21] are based on a misquotation from [13].Clopen selectors (and stronger notions, as “canonical clopen selectors”), and thus Skulaspaces, are intensively studied by U. Abraham, R. Bonnet, W. Kubiś and M. Rubin in [4, 5,13, 3, 14, 15, 16, 17, 18] in terms of Boolean algebras, but not so much as topological spaces.Relationships between “being well-generated” and “being Skula” can be found in [13, §2.3]and in [21, Theorem 3] and is stated by (i) ⇔ (ii) in the following result. Theorem 1.
For a compact Hausdorff space X the following conditions are equivalent. (i) X is a Skula space. (ii) The Boolean algebra
Clop( X ) of clopen subsets of X is well-generated. (iii) X embeds into a compact join semilattice H ( X ) such that the join operation is contin-uous and the set of principal ideals of H ( X ) is a clopen selector for H ( X ) . This result will be proved in §2, and to show the part (i) ⇒ (iii), we use Priestley spaces andtheir hyperspaces that we introduce now.A Priestley space is a compact space having a partial order h X, ≤i with the following sepa-ration property: • For every x, y ∈ X : if x y there exists a clopen initial subset V of X such that x ∈ V and y / ∈ V .Therefore any Priestley space is -dimensional.The notion of Priestley space was introduced by Hilary Ann Priestley [40] in 1970 to extendthe duality in Boolean algebras to distributive lattices. For development of this notion, seethe books of B. A. Davey and H. A. Priestley [20, Ch. 11]; G. Gierz, K. Hofmann, K. Keimel,J. Lawson, M. Mislove and D. Scott [22, 23] and S. Roman [42, Ch. 10]. TriviallyAny compact -dimensional space can be regarded as a Priestley space, where thepartial ordering is the equality.Any Skula space is Priestley since any clopen selector separates the points.In a Priestley space, if ≤ is understood from the context, then we omit it. Now we shalldiscuss the notions of hyperspace over a Priestley space. The notion of hyperspace, over acompact space, was introduces by Leopold Vietoris [46] in 1922. For a detailed presentation of IETORIS HYPERSPACES OVER PRIESTLEY SPACES . . . (July 15th, 2020) 5 hyperspace, we refer to E. Michael [34], the books of I. Illanes and S. Nadler [27], the section“The exponential” in the book of J. D. Monk [35, Ch. 1], or in the handbook of K. P. Hart,J. Nagata and J.E. Vaughan [26, §b-6 Hyperspaces]. We can see the hyperspace H ( X ) over aPriestley space X as a free join-semilattice over X . For a general definition of free objects werefer to [33].Let h X, ≤i be a Priestley space. We define its (Vietoris) hyperspace H ( X ) as follows. • H ( X ) is the set of all nonempty closed initial subsets of h X, ≤i . • For
F, G ∈ H ( X ) , we set F ≤ G if and only if F ⊆ G . • The topology T on H ( X ) is the topology generated by the sets U + := { K ∈ H ( X ) : K ⊆ U } and V − := { K ∈ H ( X ) : K ∩ V = ∅} where U and V are any clopen initial subsets and clopen final subsets in X , respectively.Note that for any clopen initial subset U , setting V = K \ U , we have H ( X ) \ U + = V − .If ≤ is the equality on X , then H ( X ) , denoted by exp( X ) , is the set of all nonempty compactsubsets of X , and is the well-known Vietoris hyperspace.Section 2 is devoted to the proof of the following result that implies Part (iii) ⇒ (vi) inTheorem 2.1. Theorem 2. If X be a Priestley space then (1) H ( X ) is a Priestley space (Theorem ). (2) h F, G i 7→ F ∨ G := F ∪ G is a continuous semilattice operation on H ( X ) . (3) X is topologically embeddable in H ( X ) by the increasing continuous map η : x
7→ ↓ x .Moreover if X be a Skula space then X is Priestley and (4) Any (nonempty) closed initial subset K of X , i.e. a member of H ( X ) , is a finite unionof clopen principals ideals of X . In particular K is clopen (Theorem ). (5) H ( X ) is a Skula space. (6) The family { K + : K ∈ H ( X ) } is a clopen selector for H ( X ) . Let us recall that a topological space X is scattered if each nonempty subspace A ⊆ X hasan isolated point for the induced topology.For a given Skula space, to introduce its (Cantor-Bendixson) height ht CB ( X ) and its (well-founded) rank rk WF ( X ) , we need the following easy observation: for completeness we re-provethis fact.Fact 1. Let U be a clopen selector of a Skula space for X . Then (1) h U , ⊆i is well-founded: see [13, Proposition 2.15(a)] . (2) Any minimal element U x of h U , ⊆i is of the form { x } with x ∈ X : see the proof ofProposition in [13] . TARAS BANAKH, ROBERT BONNET, WIESŁAW KUBIŚ (JULY 15TH, 2020) (3) X is a scattered space: see [13, Proposition 2.7(b)] .Proof. (1) Suppose by contradiction that U x % U x % U x % · · · is a strictly decreasingsequence of members of U . Let F = T n U x n . For every y ∈ F , U y ⊆ F , and thus S y ∈ F U y = F . Since each U y is open, F is open. Hence each U x i \ F is closed and nonempty, and T i ∈ ω ( U x i \ F ) = ∅ . A contradiction.(2) Let U x be a minimal element of U . We claim that U x = { x } . Otherwise there is y ∈ U x such that y = x . By the definition, x U y and thus U y ⊆ U x . Therefore U x is not a minimalelement of U . A contradiction.(3) By Part (1), X has isolated points. Next let Y be a nonempty subspace of X . Then V := { U y ∩ Y : y ∈ Y } is a clopen selector for Y and thus Y has isolated points. (cid:3) Now we introduce the ordinal invariants of Priestley and Skula spaces. For a subspace A ⊆ X of a topological space X denote by A [1] the set of all non-isolated points in X . Put X [0] = X and for every ordinal α > define its α -th (Cantor-Bendixson) derivative X [ α ] bythe recursive formula: X [ α ] = T β<α ( X [ β ] ) [1] . It is easy to see that a topological space X is scattered if and only if X [ ρ ] = ∅ for someordinal ρ . In this case X = S α ≤ ρ (cid:0) X [ α ] \ X [ α +1] (cid:1) . Therefore for each x ∈ X we define the (Cantor-Bendixson) height , denoted by ht CBX ( x ) or more simply by ht CB ( x ) , of x in X by theformula: ht CBX ( x ) = α if and only if x ∈ X [ α ] \ X [ α +1] . For instance ht CBX ( x ) = 0 if and only if x is isolated in X . The ordinal ht CB ( X ) = sup x ∈ X ht CBX ( x ) will be called the (Cantor-Bendixson) height of X . For example ht CB ( ω β + 1) = β for anyordinal β ≥ . It follows that for a compact Hausdorff space X the ht CB ( X ) -th derived set Endpt( X ) := X [ht CB ( X )] is finite and nonempty, and Endpt( X ) is called the set of end-points of X . A scattered topological space X is called unitary if the set X [ht CB ( X )] is a singleton,denoted by { endpt( X ) } and the element endpt( X ) is called the end-point of X .Next let h W, ≤i a nonempty well founded poset. Therefore h W, ≤i has a rank function rk WF W : W → Ord defined by rk WF W ( p ) = sup( { rk WF W ( q ) + 1 : q < p } . This ordinal, also denoted by rk WF ( p ) , is called the (well-founded) rank of p in W . For instance rk WF W ( p ) = 0 if and only if p ∈ Min( W ) , i.e. p is a minimal element of W . The range of rk WF , namely sup( { rk WF W ( p ) + 1 : p ∈ W } ) is called the rank of W and is denoted rk WF ( W ) .Therefore rk WF W ( p ) < rk WF ( W ) for every p ∈ W . IETORIS HYPERSPACES OVER PRIESTLEY SPACES . . . (July 15th, 2020) 7
Note that if ρ : W → Ord is a strictly decreasing function then rk WF W ( p ) ≤ ρ ( p ) for p ∈ W and rk WF W “leaves no gap”: if γ < rk WF W ( p ) then there is q ∈ W such that q < p and rk WF W ( q ) = γ .Now assume that X is a Skula space. By Theorem 2(5), H ( X ) is also Skula and thus byFact 1(1), the posets h X ) , ≤i and h H ( X ) , ⊆i are well-founded.In Section 3, we show the following result. Theorem 3.
Let X be a Skula space and let U be a clopen selector for X . Then (1) ht CB ( X ) ≤ rk WF ( X ) < ω ht CB ( X )+1 (Theorem ). (2) rk WF ( H ( X )) ≤ ω rk WF ( X ) (Theorem ). We introduce two stronger notions of Skula spaces developped in Section 4.We say that X is a canonical Skula space whenever X has a clopen selector U := { U x : x ∈ X } such that for every x ∈ X the subspace U x is unitary and endpt( U x ) = x .Next we say that X is a tree-like canonical Skula space , ot more simply a tree-like Skulaspace whenever X has a canonical clopen selector U := { U x : x ∈ X } satisfying: for x, y ∈ X , U x and U y are either comparable or disjoint.By the definitions:tree-like Skula (1) = ⇒ canonical Skula (2) = ⇒ Skula (3) = ⇒ Scattered . None implication is reversible. For (1): take an uncountable almost disjoint A on ω andconsider its Mrówka space Ψ( A ) (see below for a precise definition). For (2) see [13, Theorem3.25(a)] and for (3), see [13, Theorem 3.4] or [21, Example 1]. Note that any scattered andcompact space of (Cantor-Bendixson) height is a canonical Skula space (Proposition 3.3(a)in [13]). (cid:4) First we develop some results on canonical Skula spaces.Fact 2.
Let X be a Skula space and U be a clopen selector for X . The following are equivalent. (i) U is a canonical clopen selector for X . (ii) For any x ∈ X the subspace U x is unitary and rk WF X ( x ) = ht CBX ( x ) .Therefore the well-founded rank rk WF X ( x ) of x ∈ X does not depend of the choice of canonicalclopen selector for X . In particular rk WF ( X ) = ht CB ( X ) .Proof. Note that for every z ∈ X we have rk WF X ( z ) = rk WF U z ( z ) and ht CBX ( z ) = ht CBU z ( z ) .We show (i) implies (ii) by induction on the (well-founded) rank. If rk WF U x ( x ) = 0 then x is minimal and U x = { x } and thus ht CBU x ( x ) = 0 . Next assume that rk WF X ( y ) = ht CBX ( y ) forevery y < x . Since U x is unitary and endpt( U x ) = x rk WF X ( x ) = sup { (rk WF X ( y )+1) : y ∈ U x \{ x }} = sup { (ht CBX ( y )+1) : y ∈ U x \{ x }} = ht CBX ( x ) . TARAS BANAKH, ROBERT BONNET, WIESŁAW KUBIŚ (JULY 15TH, 2020)
Therefore rk WF X ( x ) = ht CBX ( x ) .We prove (ii) implies (i) by induction on ρ ( x ) := rk WF X ( x ) = rk WF ( U x ) . If ρ ( x ) = 0 then U x is finite and unitary, and thus U x = { x } . Next fix x ∈ X . Suppose that endpt( U y ) = y for every y < x . Since ht CB ( U y ) = ht CBX ( y ) = rk WF X ( x ) = ρ ( y ) < ρ ( x ) and U y is unitary forevery y < x , we have S y Theorem 4. If X is a canonical Skula space then its hyperspace H ( X ) is canonically Skula.Moreover, by Computation Rules , we can explicite the Cantor-Bendixson height of anymember of H ( X ) . Example 3. For instance let U σ := S x ∈ σ U x where σ = { x , x , x , x , x , x , x , x } is an an-tichain of X satisfying ht CBX ( x ) = 0 ht CBX ( x ) = 1 = ht CBX ( x ) ht CBX ( x ) = 2ht CBX ( x ) = 10 = ht CBX ( x ) ht CBX ( x ) = ω + 7 ht CBX ( x ) = 3 .By Computation Rules 4.5, we have: ht CBH ( X ) ( U σ ) = ω ω +7 + ω · ω + ω + 2 . (cid:4) The following result is proved in the same Section 4: Theorem 5. For any Skula space, the following are equivalent. (i) X is a tree-like canonical Skula space. (ii) X is a continuous image of some successor ordinal space β + 1 . In §4.1 we present some examples, namely spaces of initial subsets of a poset, Mrówka spaces(defined by almost disjoint families), Lusin families, and ladder sequences.The first examples of Skula spaces come from posets: let P be a partial ordering. We denoteby IS( P ) the set of all initial subsets of P (so ∅ , P ∈ IS( P ) ). Since IS( P ⊆ { , } P , we endow IS( P ) with the pointwise topology. We claim that IS( P ) is a Priestley space in the pointwise topology T p .Indeed, obviously IS( P ) compact. Next let x, y ∈ IS( P ) be such that x y . Pick t ∈ x \ y ⊆ P .Then V t := { z ∈ IS( P ) : t z } is a clopen initial subset of IS( P ) such that y ∈ V t and x V t .(1) In Section we characterize the posets P such that IS( P ) are Skula (such P ’s arenarrow and order-scattered [5, Theorem 1.3] ).We give examples of posets P such that IS( P ) are Skula and are (or are not) tree-likecanonically Skula. Next recall that a family A of infinite subsets of a set S is called almost disjoint if A ∩ B is finite for any distinct sets A, B ∈ A . To eliminate some triviality, we assume that A is IETORIS HYPERSPACES OVER PRIESTLEY SPACES . . . (July 15th, 2020) 9 infinite and that S A = S . A typical example of such a A is the uncountable set of branchesof a tree of height ω .A Mrówka space , also called a Ψ -space , is a unitary space of height 2. We describe a Mrówkaspace as follows. Given an infinite almost disjoint family A of subsets of a fixed set S , wedefine the compact space K A = S ∪ A ∪ {∞} as follows: all points of S are isolated, a basicneighborhood of A ∈ A is of the form V A,F := { A } ∪ ( A \ F ) with F ⊆ A finite, and a basicneighborhood of ∞ is of the form V S := K A \ S A ∈ S V A, ∅ where S is a finite subset of A . Inother words, K A is the space of ultrafilters of the Boolean subalgebra B A of P ( S ) generatedby A ∪ {{ x } : x ∈ S } which is well-known under the name of almost disjoint algebra over A .Spaces of the form K A appear often in the literature [36], although actually they wereintroduced by Alexandrov and Urysohn, and they are well-known in Set-Theoretic Topology:see [31, Ch. 3, §11]. Remark that (cid:8) { x } : x ∈ S (cid:9) ∪ A ∪ { S } is a canonical clopen selector for K A .(2) K A is a canonical Mrówka space and, by Theorem , ht CB ( H ( K A )) = ω .Therefore H ( K A ) is far from being a Mrówka space (because ht CB ( H ( K A )) ≥ ).However, by Theorem and Fact , H ( K A ) has a structure of canonically Skulaspace. On the other hand, a Mrówka space K A is embeddable in a Mrówka space G ( K A ) with acontinuous join operation (Theorem 4.12). In particular G ( K A ) is unitary, of height 2, andthus of the form K A ⋆ (where A ⋆ is an almost disjoint family). The above Item (2) can beapplied to “Lusin families” as constructed in [31, Ch. 3, Theorem 4.1], and to “ladder system”as defined in Abraham and Shelah [7].(3) There is a Lusin family L such that K L has a structure of continuous join operation.There is a ladder system L such that K L has a structure of continuous join operation. Priestley spaces and Vietoris hyperspaces In this section we show Theorem 1, that we recall under the following form. Theorem 2.1. For a compact Hausdorff space X the following conditions are equivalent. (i) X is a Skula space. (ii) X admits a partial order with clopen principal ideals. (iii) X admits a closed partial order (as subset of X × X ) with open principal ideals. (iv) The Boolean algebra Clop( X ) of clopen subsets of X is generated by a well-foundedsublattice W . (v) The Boolean algebra Clop( X ) of clopen subsets of X is generated by a clopen selector. (vi) X embeds into a compact semilattice H ( X ) with clopen principal ideals and the joinoperation on H ( X ) is continuous. Proof. The equivalence (i) ⇔ (v) was proved by Dow and Watson in [21, Theorem 3], and(iv) ⇔ (v) ⇔ (i) by Bonnet and Rubin [13, Lemma 2.8 and Proposition 2.15].To see that (ii) ⇒ (v), set U x = ↓ x := { y ∈ X : y ≤ x } for x ∈ X , and observe that U := { U x : x ∈ X } is a clopen selector. The fact that the Boolean algebra Clop( X ) is gener-ated by the family U is proved in [13, Proposition 2.15(b)].To see that (v) ⇒ (ii), fix a clopen selector { U x : x ∈ X } and observe that it induces a partialorder ≤ with clopen principal ideals defined by x ≤ y iff U x ⊆ U y .To see that (iii) ⇒ (ii), observe that for x ∈ X , since ≤ is closed in X × X , the set ↓ x := { y ∈ Y : y ≤ x } is closed in X .To see that (ii) ⇒ (iii), it suffices to prove that each partial order ≤ on X with clopen principalideals is closed in X × X . Let h x, y i ∈ ( X × X ) \ ≤ . Then W := ↓ y (that does not contains x )and V := ↓ x \ ↓ y are open in X , and thus V × W is a neighborhood of h x, y i , disjoint from thepartial order relation ≤ .The implication (vi) ⇒ (iii) is trivial because the partial order ≤ on H ( X ) is closed.The final implication (iii) ⇒ (vi) follows from Theorem 2.5. (cid:3) To prove the part (iii) ⇒ (vi), we use Priestley spaces and their hyperspaces.This section is devoted to the proof of Theorem 2.1(iii) ⇒ (vi).By a pospace (partially ordered space) we understand a topological space with a closed partialorder ≤ (for the development of this concept, see for instance L. Nachbin [37, Ch. 1]).Note that any Priestley space, and thus any Skula space, is a partially ordered space.A. Stralka [44] showed that there is a -dimensional (non scattered) compact pospace whichis not a Priestley space. On the other hand, G. Bezhanishvili, R. Mines and P. Morandi [10,Corollary 3.9] proved that for any scattered and compact space: the space is Priestley if andonly if the partial order relation is closed.In this work we assume that any Priestley and thus Skula space, and more generallyany topological space, is compact and Haudorff . Fact 2.2. If X is a Priestley space, then the space H ( X ) is a Priestley space.Proof. We denote by V X the set of clopen initial subsets of X . Since X is Priestley, S X := { U + : U ∈ V X } separates the points of H ( X ) and thus h H ( X ) , T i is Hausdorff and -dimensional.We show that H ( X ) is compact. Let V , V ⊆ V X . We set U := { U + : U ∈ V } and U := { H ( X ) \ U + : U ∈ V } . Assume that U := U ∪ U is a centered family.The set A := T V is a closed initial subset in X . Moreover, for every finite subset W of V we have (cid:0)T W (cid:1) + = T { V + : V ∈ W } 6 = ∅ . Since V + = ∅ iff V = ∅ , by the compactnes of X , A = ∅ , and thus A ∈ H ( X ) .We prove that A ∈ T U . By definition, A ∈ T { V + : V ∈ V } . Fix W ∈ V and suppose, bycontradiction, that A H ( X ) \ W + . The fact that A ∈ W + implies T V := A ⊆ W and thus IETORIS HYPERSPACES OVER PRIESTLEY SPACES . . . (July 15th, 2020) 11 (cid:0)T V (cid:1) \ W = ∅ . Since W is clopen and X is compact, there are V , . . . , V k ∈ V such that V ∩ · · · ∩ V k ⊆ W . This implies that V +1 ∩ · · · ∩ V + k ∩ ( H ( X ) \ W + ) = ∅ , contradicting the factthat U is centered. It follows that A ∈ H ( X ) \ W + for every W ∈ V . Finally, A ∈ T U . Wehave proved that H ( X ) is compact.Next we check the Priestley separation property of H ( X ) . Let F, G ∈ H ( X ) be such that G F . We fix x ∈ G \ F . For each y ∈ F , we have y x , and thus there is a clopen initialsubset V y of X such that y ∈ V y and x V y . By compactness, there is a finite subset σ of F such that F ⊆ V := S y ∈ σ V y and x V , i.e. F V . Therefore F ∈ U + and G U + .We have proved that H ( X ) is Priestley. (cid:3) Let X be a Priestley space. For x ∈ X and A, B ∈ H ( X ) we set η ( x ) := ↓ x and A ∨ B := A ∪ B . So η is a map from X into H ( X ) and ∨ is a join semilattice operation on H ( X ) . Theorem 2.3. Let X be a Priestley space. Then (1) H ( X ) is a Priestley space. (2) The map ∨ : H ( X ) × H ( X ) → H ( X ) is continuous. (3) The map η : X → H ( X ) is continuous and x ≤ y if and only if η ( x ) ≤ η ( y ) . (4) The join semilattice η [ X ] ∨ generated by η [ X ] in H ( X ) is topologically dense in H ( X ) .Proof. (1) is proved in Fact 2.2.(2) It is enough to notice that F ∪ G ∈ U + if and only if F ∈ U + and G ∈ U + .(3) It is obvious that η is an order-isomorphism. To see that η is continuous, it is enoughto notice that η − [ U + ] = U for any clopen initial subset U ⊆ X .(4) Recall that a nonempty basic open set in H ( X ) is of the form V := U + ∩ T i Let X and η : X → H ( X ) be as in Theorem . Then for any -dimensional compact join semilattice Y and any increasing continuous map f : X → Y thereis a unique join-preserving homomorphism and continuous function ˆ f : H ( X ) → Y such that ˆ f ◦ η = f . Proof. We show that the formula ˆ f ( A ) = sup f [ A ] for A ∈ H ( X ) , defines a continuous join-semilattice homomorphism ˆ f from h H ( X ) , ∨i into h Y, ∨i .Fix F, G ∈ H ( X ) . Then ˆ f ( F ∨ G ) := ˆ f ( F ∪ G ) = sup f [ F ∪ G ] = (sup f [ F ]) ∨ (sup f [ G ]) := ˆ f ( F ) ∨ ˆ f ( G ) . Thus ˆ f is a join-homomorphism.To prove the continuity of ˆ f we shall use the following well-known fact. Claim. Let L be a compact -dimensional space with a continuous semilattice operation ∨ , andlet W be a nonempty clopen initial subset of L . Then W is a finite union of clopen principalideals of the form ↓ m where m ∈ W .Sketch. For every x ∈ W let C x be a maximal chain in L containing x . Since W is clopen and C x is closed, C x ∩ W has a maximum c x . Now fix any maximal element c x in W . Since themap f c x : L → L defined by f c x ( t ) = t ∨ c x is continuous and since c x is maximal in W , ↓ c x := { t ∈ L : t ∨ c x = c x } = { t ∈ L : t ∨ c x ∈ W } := f − c x [ W ] is clopen in L . Finally by compactness, W is a finite union of clopen principal ideals of theform ↓ c x with c x ∈ W . (cid:4) For any clopen principal ideal V of Y of the form ↓ p with p ∈ Y , f − [ V ] is a clopen initialsubset of X . Let K ∈ H ( X ) . Then f [ K ] ⊆ V iff sup f [ K ] ≤ p . In other words K ∈ ( f − [ V ]) + ,i.e. K ⊆ f − [ V ] iff ˆ f ( K ) := sup f [ K ] ∈ ↓ p := V . Therefore ˆ f − [ V ] = ( f − [ V ]) + and this setis clopen in H ( X ) .Now this fact and the claim imply that ˆ f is continuous. (cid:3) The next result summarize the properties of the hyperspace of a compact Skula space. Theorem 2.5. Let X be a compact Skula space. Then X is a Priestley space and (1) H ( X ) is a Skula space. (2) Every closed initial subset K of X is clopen in X and K is finitely generated.More precisely K = ↓ σ K where σ K := Max( K ) ⊆ X is finite. (3) H ( X ) is the join semilattice generated by { η ( x ) : x ∈ X } . (4) { K + : K ∈ H ( X ) } is a clopen selector for H ( X ) .Proof. (2) Let U := { U x : x ∈ X } be a clopen selector for X . Let K be a nonempty compactinitial subset of X . Since if x ∈ K then U x := ↓ x is a clopen subset of X contained in K ,by the compactness of K , there is a finite subset σ K of K such that K = S z ∈ σ K U z . Then K = ↓ K = S z ∈ Max( K ) U z = ↓ Max( K ) .(3) is a consequence of Theorem 2.3 and of Proposition 2.4. IETORIS HYPERSPACES OVER PRIESTLEY SPACES . . . (July 15th, 2020) 13 (4) By Fact 2.2, H ( X ) is compact. By Parts (2) and (3) any K ∈ H ( X ) is a clopen initialsubset X and thus K + := { L ∈ H ( X ) : L ⊆ K } is clopen in H ( X ) . Now it is easy to checkthat { K + : K ∈ H ( X ) } is a clopen selector for H ( X ) .(1) follows from (4) and the fact that H ( X ) is compact. (cid:3) Interplay between (Cantor-Bendixson) height and (well-founded) rank The goal of this section is to show Theorem 3: (1): if X is a Skula space then ht CB ( X ) ≤ rk WF ( X ) < ω ht CB ( X )+1 and (2): rk WF ( H ( X )) ≤ ω rk WF ( X ) .Concerning Part (1) of Theorem 3 we prove a little bit more: Theorem 3.1. For each Skula pospace h X, ≤i we have: ht CB ( X ) ≤ rk WF ( X ) < ω ht CB ( X ) · (cid:0) | Endpt( X ) | + 1 (cid:1) < ω ht CB ( X )+1 . In this theorem ω ht CB ( X ) is the ht CB ( X ) -th exponent of the ordinal ω . The exponentiationof ordinals is defined by transfinite induction: α = 1 and α γ = sup β<γ ( α β · α ) for γ > .Recall that the multiplication of ordinals is also defined by transfinite induction: α · and α · γ = sup β<γ ( α · β + α ) for γ > .We cannot improve Theorem 3.1 using ordinal invariants. Indeed(1) For the first inequality: if X is canonically Skula then rk WF ( X ) = ht CB ( X ) (see Fact 2).(2) For the inequality rk WF ( X ) < ω ht CB ( X )+1 , consider the ordinal space X n := ω + n ( n ≥ ). Then ht CB ( X n ) = 1 < rk WF ( X n ) = ω + n and sup n rk WF ( X n ) = ω = ω ht CB ( X n )+1 .Theorem 3.1 is a consequence of the following result. Lemma 3.2. For any point x ∈ X of a Skula pospace X we have ht CBX ( x ) ≤ rk WF X ( x ) < ω ht CB X ( x ) · ( | Endpt( ↓ x ) | + 1) . Proof. The proof is divided in two parts.(1) By induction on ht CB ( x ) , we prove the inequality ht CB ( x ) ≤ rk WF ( x ) . This inequality istrivially true if ht CB ( x ) = 0 . Assume that for some ordinal α > the inequality ht CB ( y ) ≤ rk WF ( y ) has been proved for all points y ∈ X satisfying ht CB ( y ) < α . Choose any point x ∈ X with ht CB ( x ) = α . Taking into account that ↓ x is clopen and thus open, observe that ht CBX ( x ) = min W sup y ∈ W \{ x } (ht CB ( y )+1) ≤ sup y ∈↓ x \{ x } (ht CB ( y )+1) ≤ sup y ∈↓ x \{ x } (rk WF ( y )+1) := rk WF X ( x ) where the minimum, min W , is taken over all open neighborhoods W of x in X .(2) We shall prove rk WF X ( x ) < o X ( x ) := ω ht CB ( ↓ x ) · (cid:0) | Endpt( ↓ x ) | +1 (cid:1) by induction on the ordinal o X ( x ) . By the definition of o X , the map o X : h X, ≤i → h Ord , ≤i defined by x o X ( x ) is increasing.Assume first that o X ( x ) ≤ ω . Since | Endpt( ↓ x ) | + 1 ≥ we have ht CB ( ↓ x ) = 0 . Thereforethe set ↓ x is finite and hence rk WF X ( x ) ≤ |↓ x | < |↓ x | + 1 := o X ( x ) .Next suppose that for some ordinal α – of the form o . ( . ) – the inequality rk WF T ( t ) < o T ( t ) has been proved for all Skula pospaces T and all points t ∈ T satisfying o T ( t ) < α .Fix a point x ∈ X such that o X ( x ) = ω ht CB ( ↓ x ) · (cid:0) | Endpt( ↓ x ) | +1 (cid:1) := α . We set E =Endpt( ↓ x ) and, since E is a finite poset, we fix e ∈ Min( E ) . Let Y = ↓ e and Z = ( ↓ x ) \ ( ↓ e ) ( Z can be empty). So Y and Z are clopen subspaces of X , and Y is unitary and e = endpt( Y ) .Note that Y is an initial subset of ↓ x and that ht CB ( ↓ e ) = ht CB ( Y ) = ht CB ( ↓ x ) .Fix y ∈ Y with y < e . So ht CB ( ↓ y ) < ht CB ( ↓ e ) and thus ω ht CB ( ↓ y ) < ω ht CB ( ↓ e ) = ω ht CB ( ↓ x ) .Since e ∈ Min( E ) , by induction hypothesis, rk WF ( y ) < o X ( y ) := ω ht CB ( ↓ y ) ( | E ] + 1) < ω ht CB ( ↓ e ) .This fact plus the fact that Y is unitary imply:(i) rk WF ( Y ) := rk WF Y ( e ) ≤ ω ht CB X ( ↓ e ) = ω ht CB X ( ↓ x ) . Case . | E | = 1 , i.e. E = { e } . Since e ∈ Min( E ) and e Z := ( ↓ x ) \ ( ↓ e ) we have ht CB ( Z ) < ht CB ( Y ) , and thus by induction hypothesis,(ii) rk WF ( Z ) = rk WF Z ( x ) < o Z ( x ) := ω ht CB ( Z ) · ( | Endpt( Z ) | + 1) < ω ht CB ( Y ) = ω ht CB ( ↓ x ) .Since Y is an initial subset of ↓ x and thus Z is a final subset of ↓ x , using (i) and (ii), weobtain: rk WF ( ↓ x ) ≤ rk WF ( Y + Z ) ≤ rk WF ( Y ) + rk WF ( Z ) < ω ht CB ( ↓ x ) + ω ht CB ( ↓ x ) == ω ht CB ( ↓ x ) · := o X ( x ) = α where Y + Z denotes the lexicographic sum of Y and Z , and so y ′ < z ′ for every y ′ ∈ Y and z ′ ∈ Z . Case . | E | ≥ . Recall that Y := ↓ e and Z := ( ↓ x ) \ ( ↓ e ) . Since E \ { e } is nonempty, ht CB ( Y ) := ht CB ( ↓ x ) = ht CB ( Z ) and Endpt( Z ) = E \ { e } . So | Endpt( Z ) | = | E | − < | E | . Bythe induction hypothesis,(iii) rk WF ( Z ) < o Z ( x ) := ω ht CB ( Z ) · (cid:0) | Endpt( Z ) | +1 (cid:1) = ω ht CB ( ↓ x ) ·| E | .As in Case 1, by (i) and (iii), we obtain: rk WF X ( x ) = rk WF ( ↓ x ) ≤ rk WF ( Y ) + rk WF ( Z ) < ω ht CB ( ↓ x )) + ω ht CB ( ↓ x )) · | E | << ω ht CB ( ↓ x ) · (cid:0) | E | +1 (cid:1) := o X ( x ) = α , that ends the proof of the lemma. (cid:3) IETORIS HYPERSPACES OVER PRIESTLEY SPACES . . . (July 15th, 2020) 15 Proof of Theorem . By Lemma 3.2, for x ∈ X we have ht CBX ( x ) ≤ rk WF X ( x ) < ω ht CB X ( ↓ x ) · (cid:0) (cid:12)(cid:12) ( ↓ x ) [ht CB X ( ↓ x )] (cid:12)(cid:12) + 1 (cid:1) := ω ht CB X ( ↓ x ) · m << ω ht CB ( X ) · ω = ω ht CB ( X )+1 where m := (cid:12)(cid:12) ( ↓ x ) [ht CB X ( ↓ x )] (cid:12)(cid:12) + 1 < ω . Hence ht CB ( X ) ≤ rk WF ( X ) < ω ht CB ( X )+1 : the lastinequality follows from the fact that there is an x ∈ X such that rk WF ( X ) = rk WF ( x ) . (cid:3) Our next step is to prove: ht CB ( H ( X )) ≤ rk WF ( H ( X )) ≤ ω rk WF ( X ) < ω ω ht CB ( X )+1 . The first and the last inequality follows easily from Theorem 3.1. The difficult part is theinequality rk WF ( H ( X )) ≤ ω rk WF ( X ) , for which we need some preparation. Any (nonzero)ordinal α has a Cantor decomposition : α := ω α p + · · · + ω α ℓ p ℓ where α > · · · > α ℓ and p i > for i ≤ ℓ . We define the Hessenberg’s natural sum of ordinals (also called the polynomial sum )of the ordinals α := ω γ p + · · · + ω γ m p m and β := ω γ q + · · · + ω γ m q m (where the p i ’s and q i ’scan be ), as the ordinal: α ⊕ β = ω γ ( p + q ) + · · · + ω γ m ( p m + q m ) . For example if α = ω ω + ω ω and β = ω ω + ω + ω +5 then α ⊕ β = ω ω + ω ω ω + ω ω +5 .Notice that ⊕ has the following properties: for every ordinals α , β , γ and δ we have(i) α ⊕ β = β ⊕ α .(ii) ( α ⊕ β ) ⊕ γ = α ⊕ ( β ⊕ γ ) .(iii) α ⊕ α .(iv) β < γ if and only if α ⊕ β < α ⊕ γ .(v) α, β < ω δ implies α ⊕ β < ω δ .In [1, Item (1), p. 55] it is also shown that for every α , β and γ :(vi) α ⊕ β is strictly increasing in both arguments.(vii) if γ < α ⊕ β then there are α ′ ≤ α and β ′ ≤ β such that γ = α ′ ⊕ β ′ (with α ′ < α or β ′ < β ). (Do not be tempted to think that if α < γ < α ⊕ β then γ = α ⊕ β ′ for some β < β .)We give a useful application of the ⊕ operation due to Telgàsky [45, Theorem 2]. A proof canalso be found in Pierce [38, Ch. 21, Proposition 2.21.1]. Theorem 3.3 (Telgàsky) . Let X and Y be compact scattered spaces. For every h x, y i ∈ X × Y we have: ht CBX × Y ( h x, y i ) = ht CBX ( x ) ⊕ ht CBY ( y ) , and thus ht CB ( X × Y ) = ht CB ( X ) ⊕ ht CB ( Y ) .Moreover if X and Y are unitary then X × Y is unitary and endpt( X × Y ) = h endpt( X ) , endpt( Y ) i . (cid:3) Let h W, ≤i be a nonempty well founded poset. We denote by [ W ] <ω ∗ the set of nonemptyfinite subsets of W and by K ( W ) be the set of all initial subsets of W generated by a nonemptyfinite subset W . That is,( ∗ ) I ∈ K ( W ) if and only if there is σ ∈ [ W ] <ω ∗ such that I = ↓ σ := S p ∈ σ ↓ p .Moreover we may assume that σ is an antichain of W , and thus σ = Max( I ) .By a result of Birkhoff [11, Ch. VIII, §2, Theorem 2], h K ( W ) , ⊆i is well-founded. Thereforethe rank function rk WF K ( W ) : K ( W ) → Ord is well-defined.The next result is due to N. Zaguia [47, Ch. 1, Theorem II-1.2] (Thesis, in French, 1983).For completeness we recall his proof. Theorem 3.4 (Zaguia) . Let h W, ≤i be a well founded poset. Then rk WF ( K ( W )) ≤ ω rk WF ( W ) .Proof. By the definition, rk WF K ( W ) ( I ) := sup( { rk WF K ( W ) ( J ) + 1 : J ∈ K ( W ) and J $ I } ) for any I ∈ K ( W ) . For instance rk WF K ( W ) ( I ) = 0 if and only if there is a minimal element p of W such that I = { p } . To prove the theorem, we need some preliminary facts and the mainkey is the following result. Claim . Let I ∈ K ( W ) , and let I ′ , I ′′ ∈ K ( W ) be such that I = I ′ ∪ I ′′ . Then rk WF K ( W ) ( I ) ≤ rk WF K ( W ) ( I ′ ) ⊕ rk WF K ( W ) ( I ′′ ) .Proof. The proof is done by induction on β := rk WF K ( W ) ( I ) . First if β = 0 , i.e. I = { p } with p ∈ Min( W ) , there is nothing to prove.Next let I, I ′ , I ′′ ∈ K ( W ) be such that I = I ′ ∪ I ′′ with rk WF K ( W ) ( I ) = β . Let J ∈ K ( W ) be such that J $ I . Setting J ′ := J ∩ I ′ and J ′′ := J ∩ I ′′ we have (1): J = J ′ ∪ J ′′ , (2): J ′ ⊆ I ′ and J ′′ ⊆ I ′′ and (3): J ′ = I ′ or J ′′ = I ′′ . Since rk WF K ( W ) ( J ) < β := rk WF K ( W ) ( I ) , by theinduction hypothesis we have rk WF K ( W ) ( J ) ≤ rk WF K ( W ) ( J ′ ) ⊕ rk WF K ( W ) ( J ′′ ) . Since J ′ = I ′ or J ′′ = I ′′ , rk WF K ( W ) ( J ′ ) ⊕ rk WF K ( W ) ( J ′′ ) < rk WF K ( W ) ( I ′ ) ⊕ rk WF K ( W ) ( I ′′ ) . Hence rk WF K ( W ) ( J ) < rk WF K ( W ) ( I ′ ) ⊕ rk WF K ( W ) ( I ′′ ) and thus rk WF K ( W ) ( I ) ≤ rk WF K ( W ) ( I ′ ) ⊕ rk WF K ( W ) ( I ′′ ) . (cid:4) Claim . For every I ∈ K ( W )rk WF K ( W ) ( I ) ≤ rk WF K ( W ) ( ↓ p ) ⊕ · · · ⊕ rk WF K ( W ) ( ↓ p n − ) where { p , . . . , p n − } := Max( I ) .Proof. Claim 2 follows from Claim 1 applied to I = S i Proof. By induction on β := rk WF W ( p ) . First assume β = 0 . So p ∈ Min( W ) and thus { p } = ↓ p ∈ Min( K ( W )) . Hence rk WF K ( W ) ( ↓ p ) = 0 , and consequently rk WF K ( W ) ( ↓ p ) = 0 < ω = ω rk WF W ( p ) .Now let p ∈ W be such that β = rk WF W ( p ) . Let I ∈ K ( W ) be such that I $ ↓ x . Let { p , . . . , p n − } := Max( I ) . For each p i < p we have rk WF W ( p i ) < β := rk WF W ( p ) , and thus bythe induction hypothesis rk WF K ( W ) ( ↓ p i ) < ω rk WF W ( p i ) < ω rk WF W ( p ) := ω β . Since I = S i Let X be a Skula space. Fix a clopen selector U = { U x : x ∈ X } for X .So U defines a well-founded poset h X, ≤i where we set x ≤ y if and only is U x ⊆ U y . ByTheorem 2.5(2), a member G of H ( X ) is an initial subset of h X, ≤i generated by a nonemptyfinite subset of X . In other words K ( X ) = H ( X ) is Skula and h H ( X ) , ⊆i = h K ( X ) , ≤i .Therefore by Zaguia Theorem 3.4 rk WF ( H ( X )) ≤ ω rk WF ( X ) . (cid:4) Now we can state the second result on the relationship between the rank and the heightfunctions. Theorem 3.6. For any compact Skula pospace X , its hyperspace H ( X ) satisfies: ht CB ( H ( X )) ≤ rk WF ( H ( X )) ≤ ω rk WF ( X ) < ω ω ht CB ( X )+1 . Proof. By Theorem 3.1, ht CB ( H ( X )) ≤ rk WF ( H ( X )) , and rk WF ( X ) < ω ht CB ( X )+1 and thus ω rk WF ( X ) < ω ω ht CB ( X )+1 . Finally the inequality rk WF ( H ( X )) ≤ ω rk WF ( X ) is proved in Re-mark 3.5. (cid:3) Note that the above inequality rk WF ( H ( X )) ≤ ω rk WF ( X ) can not be improved, using topo-logical cardinal functions: see Theorem 4.7 on scattered compact and hereditary paracompactspaces.4. Canonical and tree-like Skula spaces and Applications to some classes Our first goal is to prove Theorem 4 that we restate in Theorem 4.1. It is a consequence ofPropositions 4.3 and Theorem 4.4. We use only the following facts of a clopen selector U for X : • Any closed initial subset of X is a finite union of members of U , and thus is clopen.In particular, • The intersection of any two members of U is a finite union of members of U .We introduce some notation. Given a set I let [ I ] <ω be the set of finite subsets of I and [ I ] <ω ∗ = [ I ] <ω \ {∅} . Theorem 4.1. Let X be a canonical Skula space. Then the space H ( X ) is a canonical Skulaspace. More precisely { U + : U ∈ H ( X ) } is a canonical clopen selector for H ( X ) . To prove the theorem, we need some preliminary facts. We fix a canonical selector U := { U x : x ∈ X } for the Skula space X . So U x := ↓ x . Also in what follows a “finite and nonempty antichain”of X is abbreviated by “antichain”.For a clopen selector U of a Skula space X , K ∈ H ( X ) and x ∈ X , ht CB ( K ) denotes the height of the space K (as compact subspace of X ), ht CBH ( X ) ( K ) denotes the height of K as element of H ( X ) , and rk WF X ( x ) denotes the rank of x as element of X , . . . Lemma 4.2. Let X be a canonical Skula space. Let x ∈ X and σ be an antichain of X satisfying x U σ . Then ht CB ( U x \ U σ ) = ht CB ( U x ) .Proof. There is an antichain τ ⊆ U x such that U x ∩ U σ = U τ . Hence U z $ U x for z ∈ τ . Since U x is unitary with end-point x and τ is finite with x U τ , we have ht CB ( U τ ) < ht CB ( U x ) .Hence ht CB ( U x \ U σ ) = ht CB ( U x ) . (cid:3) Proposition 4.3. Let X be a canonical Skula space, and let σ be an antichain of X such that | σ | ≥ . Then ht CBH ( X ) ( U σ ) = L x ∈ σ ht CBH ( X ) ( U x ) .Proof. Let h x i i i We remark that h K ∩ U ′ x i x ∈ σ depends of the enumeration h x i i i Let U := { U x : x ∈ X } be a canonical selector for X and let x ∈ X . Then rk WF X ( x ) = ht CBX ( x ) = ht CB ( U x ) and (1) If rk WF X ( x ) = 0 then ht CBH ( X ) ( U x ) = 0 . (2) If rk WF X ( x ) = 1 then ht CBH ( X ) ( U x ) = 1 . (3) If rk WF X ( x ) = 1+ α ≥ then ht CBH ( X ) ( U x ) = ω α .Moreover for any x ∈ X the subspace H ( U x ) = U + x of H ( X ) is unitary and endpt( H ( U x )) = U x .Proof. Since each U x is unitary with end-point x , by Fact 2, rk WF X ( x ) = ht CBX ( x ) = ht CB ( U x ) .So we compute only ht CBH ( X ) ( U x ) by induction on rk WF X ( x ) = ht CBX ( x ) .We need to recall that the Hessenberg product ⊙ of ordinals is defined as follows (see [2]) .Let α and β be ordinals. We set α ⊙ α , α ⊙ ( β + 1) = ( α ⊙ β ) ⊕ β and α ⊙ β = sup γ<β α ⊙ γ for a limit β .Operation ⊙ is not the same as the Hessenberg multiplication α ⊗ β which is obtained fromthe normal forms of α and β viewed as polynomials and multiplied accordingly. In particular, α ⊗ β is commutative, but ⊙ is not. For instance ω ⊗ ⊗ ω = ω + ωω ⊙ ω ⊕ ω = ω + ω and ⊙ ω = sup n<ω ⊙ n = ω .The function α ⊙ β is strictly increasing in the right variable, continuous in the right variable,and non-decreasing in the left variable. ObviouslyIf n ∈ ω then ω α · n = ω α ⊙ n = ω α ⊗ n .It is easy to check that α · β ≤ α ⊙ β ≤ α ⊗ β where α · β is the usual operation. Case . rk WF X ( x ) = 0 . So U x = { x } and thus ht CBH ( X ) ( U x ) = 0 . Obviously H ( U x ) = { x } andthus H ( U x ) is unitary with endpt( H ( U x )) = { x } = U x . Case . rk WF X ( x ) = 1 . The set U x \ { x } is infinite and discrete. Recall that H ( U x ) is theset of all nonempty final subsets of U x , considered as subsepace of X . It is easy to see thatthe set of all nonempty antichains contained in U x ∩ Max( X ) is the set of isolated points of H ( U x ) . Hence H ( U x ) is homeomorphic to the one-point compactification of a discrete spaceand endpt( H ( U x )) = U x . Therefore ht CBH ( X ) ( U x ) = 1 . Case . rk WF X ( x ) = 1 + α with α = 1 . The set S := { y ∈ X : y < x and rk WF X ( y ) = 1 } = { y ∈ U x \ { x } : rk WF X ( y ) = 1 } is infinite. In particular if y < x and y S then rk WF X ( y ) = 0 . Let σ be an antichain of U x such that x σ . Since S y ∈ σ U y := U σ ⊆ U x , by Proposition 4.3, ht CBH ( X ) ( U σ ) = L y ∈ σ ∩ S ht CBH ( X ) ( U y ) ⊕ L y ∈ σ \ S ht CBH ( X ) ( U y )= 1 ⊙ | σ ∩ S | ⊕ | σ ∩ S | . In particular if σ is an antichain contained in S then ht CBH ( X ) ( U σ ) = L y ∈ σ ht CBH ( X ) ( U y ) = 1 ⊙ | σ | = | σ | . Now, since S is infinite, it is easy to see that ht CBH ( X ) ( U x ) = ω := ω . Obviously H ( U x ) isunitary and endpt( H ( U x )) = U x . Case . rk WF X ( x ) = 1 + α with α ≥ . Fix β < α with β ≥ . The set S β = { y ∈ X : y < x and rk WF X ( y ) = β } = { y ∈ U x \ { x } : rk WF X ( y ) = β } is infinite. Now let σ be an antichain contained in S β and U σ := S y ∈ σ U y . So U σ ⊆ U x \ { x } .Remark that for every y ∈ σβ = rk WF X ( y ) ≤ γ := max { rk WF X ( z ) : z ∈ σ } < α .For every y ∈ S β , we have β = rk WF X ( y ) ≤ γ < α . Moreover by Fact 2, rk WF X ( y ) =ht CBX ( y ) = ht CB ( U y ) . In other words, β = ht CB ( U y ) ≤ γ .Fix y ∈ S β . Since β ≥ , by the induction hypotheses, we have ht CBH ( X ) ( U y ) = ω β .(Note that if β = α = 2 then by Case 3, ht CBH ( X ) ( U y ) = ω := ω = ω β .) As in Case 3, byProposition 4.3, ht CBH ( X ) ( U σ ) = L y ∈ σ ht CBH ( X ) ( U y ) = ω β ⊙ | σ | = ω β · | σ | < ω α .Now since S β is infinite, β is any ordinal (strictly) less than α and | σ | is arbitrary, it is easyto check that ht CBH ( X ) ( U x ) = sup { ω β · m : β < α and m ∈ ω } = ω α .So H ( U x ) is unitary and endpt( H ( U x )) = U x because ht CBH ( X ) ( U σ ) < ω α = ht CBH ( X ) ( U x ) forany finite antichain σ of U x satisfying U σ $ U x .We have proved Theorem 4.4. (cid:3) Proof of Theorem . By Theorem 2.5(4), the set of U + := { K ∈ H ( X ) : K ⊆ U } , where U is any clopen initial subset of X , defines a clopen selector for H ( X ) . Therefore it suffices toprove that each U + is unitary and clopen subset of H ( X ) with endpt( U + ) = U .We fix closed (equivalently clopen) initial subsets U and V of X . So U = U σ and V = U ρ where σ and ρ are antichains of X . Assume that U ρ $ U σ . It suffices to show that( ⋆ ) ht CB ( U ρ ) < ht CB ( U σ ) .We show ( ⋆ ) by induction on | σ | . If | σ | = 1 , then we are done by Theorem 4.4. Next assumethat | σ | ≥ . Notice that U ρ = U ρ ∩ U σ = S s ∈ σ ( U ρ ∩ U s ) and that, by compactness, each IETORIS HYPERSPACES OVER PRIESTLEY SPACES . . . (July 15th, 2020) 21 U ρ ∩ U s is a finite union of U z ’s. For any s ∈ σ we have U ρ ∩ U s ⊆ U s . By Theorem 4.4, ht CB ( U ρ ∩ U s ) ≤ ht CB ( U s ) for any s ∈ X .Since U ρ $ U σ , there is s ′ ∈ σ such that U ρ ∩ U s ′ $ U s ′ . Again, by Theorem 4.4, ht CB ( U ρ ∩ U s ′ ) < ht CB ( U s ′ ) . But the function h α, β i 7→ α ⊕ β is strictly increasing in both arguments.Therefore by Proposition 4.3, ht CB ( U ρ ) = L s ∈ σ ht CB ( U ρ ∩ U s ) < L s ∈ σ ht CB ( U s ) = ht CB ( U σ ) .This ends the proof of Theorem 4.1. (cid:3) Computation Rules 4.5. For a nonempty antichain σ of X , by Theorem 4.1, the subspace H ( U σ ) = U σ + of H ( X ) is unitary with endpt( H ( U σ )) = U σ , and we can calculate ht CBH ( X ) ( U σ ) :- by Propositions 4.3 we have ht CBH ( X ) ( U σ ) = L x ∈ σ ht CBH ( X ) ( U x ) , and- by Theorem 4.4 we know ht CBH ( X ) ( U x ) in function of ht CBX ( x ) for any x ∈ X .Such a calculation appears in Example 3. (cid:4) Next we develop properties of tree-like canonical Skula spaces: Proposition 4.6, (i) ⇔ (ii) wasproved U. Abraham and R. Bonnet [1] and (ii) ⇔ (iii) is due to R. Bonnet and H. SiKaddour[12, §2.4 and §2.6].The “Moreover” is due to M. Rubin: we recall that a topological space Z is retractable whenever every nonempty closed set F of Z is a retract, i.e. there is a continuous map f : Z → F such that f ↾ F is the identity on F .For completeness we give a sketch of the proof of the next result, simplifying the techonologyof the original proof. Proposition 4.6. Let X be a compact space. The following are equivalent. (i) X is a scattered space and X is a continuous image of a -dimensional complete linearordered space h X, ≤i endowed with the order topology. (ii) X is a continuous image of a successor ordinal α + 1 (endowed with the order topology). (iii) X has a tree-like canonical clopen selector.Moreover if X is a continuous image of a successor ordinal then X is retractable.Hint. Let us begin by an observation. Let α be an infinite ordinal. Then { [0 , β ] : β ≤ α } is aclopen selector for [0 , α ] .On the other hand, as in the proof of Theorem 3.1, any non-zero ordinal β has a Cantordecomposition: β := ω β p + · · · + ω β ℓ p ℓ where β > · · · > β ℓ and p i ≥ for i ≤ ℓ . Denote by tip( β ) the last block. For instance, if β = ω ω · ω + ω · then tip( β ) = ω and if β = ω + 5 then tip( β ) = 1 . So [tip( β ) , β ) is order-isomorphic to tip( β ) := ω β ℓ . Therefore tip( β ) + 1 andthus U β := (tip( β ) , β ] are unitary and of Cantor-Bendixson height β ℓ . Now, obviously the setof (cid:8) U β : β ≤ α (cid:9) is a tree-like canonical clopen selector for [0 , α ] . (ii) ⇒ (iii) [12, §2.6]. The space X is a continuous image of α + 1 , which means that theBoolean algebra Clop( X ) is a (superatomic) subalgebra of Clop([0 , α ]) . We recall this construc-tion of the tree-like canonical selector U for X . For any U ∈ Clop( X ) there is a unique finitestrictly increasing sequence ~s U := h s Ui i i< ℓ ( U ) of members of α +1 such that: U = S i<ℓ ( s U i , s U i +1 ] with ( s U i +1 , s U i +2 ] = ∅ for i < ℓ ( U ) − . Fix x ∈ X . Let m [ x ] = min (cid:8) ℓ ( U ) ∈ ω : U ∈ Clop( X ) and Endpt( U ) = { x } (cid:9) , and U [ x ] = (cid:8) U ∈ Clop( X ) : Endpt( U ) = { x } and ℓ ( U ) = m [ x ] (cid:9) . We recall that for any integer n (in particulare if n := m [ x ] ), the set [ α + 1] n of finite strictlyincreasing sequences of α + 1 of length n is well-ordered by the lexicographic order relationdenoted by (cid:22) . Hence S [ x ] := { ~s U : U ∈ U [ x ] } is well-ordered by (cid:22) , and thus U x := min( U [ x ]) exists.Therefore U := { U x : x ∈ X } is the required tree-like canonical clopen selector for X : see [12,§2.6: Part C].(iii) ⇒ (ii) [12, §2.4]. We use the following fact, whose proof can be obtained by inductionon the (well-founded) rank rk WF ( X ) of X : for analogous results see S. Todorčević in [31, Ch6 §2] and S. Koppelberg [28, Ch. 6 §16]. Let U := { U x : x ∈ X } be a tree-like canonicalselector for X , considered as a well-founded set of subsets of X . So x ≤ y if U x ⊆ U y . Thereare a well-ordering (cid:22) on X and a one-to-one map ϕ : U → P ( X ) , satisfying that for every x, y ∈ X :(1) If x ≤ y then x (cid:22) y ,(2) ϕ ( U x ) is an half-open interval in h X, (cid:22)i of the form ( a x , b x ] with a x , b x ∈ X ,(3) If U x = { x } then ϕ ( U x ) is a singleton, and(4) U x ⊆ U y iff ϕ ( U x ) ⊆ ϕ ( U y ) , and U x ∩ U y = ∅ iff ϕ ( U x ) ∩ ϕ ( U y ) = ∅ .By Sikorski’s extension theorem [28, Theorem 5.5], we extend ϕ in a one-to-one Boolean mapfrom Clop( X ) into the interval algebra B ( X ) over h X, (cid:22)i .(i) ⇔ (ii). See Abraham and Bonnet [1, Theorem 1].The “Moreover” part is a re-statement of a result of M. Rubin [41, Theorem 5.1] proved interms of Boolean algebras. (cid:3) Comment. M. Pouzet [39] gave the following proof of Proposition 4.6(iii) ⇒ (i). Let X bea set and let F be a family of nonempty subsets of X such that two members of F areeither comparable or disjoint. Then F is order-isomorphic to a set of intervals of a linearordering. He proved first the case where F is finite and then made the final conclusion usingthe “Compactness Theorem”. (cid:4) IETORIS HYPERSPACES OVER PRIESTLEY SPACES . . . (July 15th, 2020) 23 We recall that a topological space X is hereditarily paracompact if each subspace of X isparacompact (and hence Hausdorff).Dow and Watson [21, Corollary 2] proved that each hereditarily paracompact scatteredcompact topological space is a Skula space. We generalize this result in the next result, using aclassification of compact hereditary paracompact spaces, proved by Banakh and Leiderman [9]. Theorem 4.7. Let X be a scattered compact and hereditary paracompact space. Then X hasa tree-like canonical clopen selector.Therefore, by Proposition ⇒ (ii) , X is a continuous image of a successor ordinal en-dowed with the order topology.Proof. According to [9, Theorem 3(3)], the class of hereditarily paracompact scattered compactspaces coincides with the smallest class A that contains all singletons and is closed underAleksandrov compactifications (also called “one-point compactifications”) of topological sumsof unitary scattered compact spaces from the class A .More precisely A = S α ∈ Ord A α where A is the class of singletons and A α is the class ofspaces which can be written as Aleksandrov compactifications of topological sums of unitaryscattered compact spaces from the class S β<α A β .Now let X be a scattered compact hereditary paracompact space. So X ∈ A α \ S β<α A β .We show that X has a tree-like selector by induction on α . If X ∈ A then there is nothing toprove, and if X is a finite topological sum of members of A then X ∈ A . Next suppose that α ≥ . Then there are an infinite family B := { B i : i ∈ I } and a point a such that(i) for every distinct i, j ∈ I : B i ∈ A β i with β i < α , and B i ∩ B j = ∅ ,(ii) the topological sum L i ∈ I B i is locally compact and not compact, and (cid:0)L i ∈ I B i (cid:1) ∪ { a } is the Aleksandrov compactification of L i ∈ I B i .By the induction hypothesis, for each i ∈ I the space B i has a partial order (cid:22) i satisfying B i = ↓ a i , i.e. y (cid:22) i a i for every y ∈ B i . We set y (cid:22) X a for every y ∈ L i ∈ I B i . Obviously, thebinary relation (cid:22) defined as (cid:22) X ∪ S i ∈ I (cid:22) i is a partial order relation on X . It is easy to checkthe following facts.(1) For x ∈ X with x = a , we have rk WF X ( x ) = ht CBX ( x ) < α and rk WF ( X ) = rk WF X ( a ) =ht CBX ( X ) = ht CB ( X ) = α .(2) For every x ∈ X , ↓ x := { y ∈ X : y (cid:22) x } is a unitary space with end-point x .(3) For distinct x, y ∈ X , x and y are incomparable if and only if ( ↓ x ) ∩ ( ↓ y ) = ∅ .Hence {↓ x : x ∈ X } is a tree-like selector for X . (cid:3) Note that the class of tree-like Skula spaces of larger than the class of hereditary paracompactand compact space. For example ω + 1 is compact but not hereditary paracompact (considerthe subspace ω := [0 , ω ) ). But ω + 1 is a tree-like Skula space. The space of initial subsets of a partial ordering. Let P be a poset. Recall that IS( P ) denotes the set of all initial subsets of P (so ∅ , P ∈ IS( P ) ). Let FS( P ) be the set ofall final subsets of P . Then ϕ : IS( P ) → FS( P ) defined by ϕ ( I ) = P \ I is an isomorphismbetween the complete distributive lattices h IS( P ) , ⊆i and h FS( P ) , ⊇i .Since IS( P ) , FS( P ) ⊆ { , } P , we endow IS( P ) and FS( P ) with the pointwise topology.Hence the spaces IS( P ) and FS( P ) are compact and ϕ is a homeomorphism onto. So:( ⋆ ) We identify the Priestley spaces h IS( P ) , ⊆i and h FS( P ) , ⊇i endowed with the pointwisetopology T p . In [5, Theorem 2.3], it is shown:( ⋆⋆ ) The Boolean algebra Clop(FS( P )) of clopen subsets of FS( P ) is the poset algebra F ( P ) .For the definition and properties of (free) poset algebras F ( P ) see [5] and [4, §3].We say that a poset P is narrow if every antichain (set of pairwise incomparable elements)is finite. A poset P is order-scattered if P does not contain a copy of the rational chain Q .The next result can be found in [5, Theorem 1.3]. Proposition 4.8. Let P be a poset. The following are equivalent. (i) P is a narrow and order-scattered poset. (ii) FS( P ) is a scattered space, i.e. the poset algebra F ( P ) is superatomic. (iii) FS( P ) is a Skula space, i.e. the poset algebra F ( P ) is well-generated. (cid:3) A poset P is a well-quasi ordering (w.q.o.) whenever P is narrow and well-founded. Thenotion of w.q.o. was introduced by G. Kurepa in 1937, cited in [30], and is a frequentlydiscovered concept: see for instance Kruskal [29]. We recall two facts for which the proof isobvious. Proposition 4.9. Let P be a partial ordering. The following are equivalent. (i) P is a well-quasi ordering. (ii) h IS( P ) , ⊆i (i.e. h FS( P ) , ⊇i ) has no strictly decresing sequence. (iii) Any nonempty final subset K of P is finitely generated, i.e. K contains a nonemptyfinite subset σ such that K = ↑ σ . (cid:3) At the opposite of Proposition 4.8(i) ⇒ (iii), for which the proof is far to being obvious, theproof in some special case is quite trivial. Proposition 4.10 (Special case of Proposition 4.8) . Let P be a well-quasi ordering. Then FS( P ) is a Skula space, and thus FS( P ) is a scattered space.Proof. Obviously FS( P ) is compact. By Proposition 4.9, for any K ∈ FS( P ) there is anonempty finite antichain σ K in P such that K = ↑ σ K . Therefore IETORIS HYPERSPACES OVER PRIESTLEY SPACES . . . (July 15th, 2020) 25 U + K := { F ∈ FS( P ) : σ F ⊆ G } = { F ∈ FS( P ) : F ⊇ K } is a clopen subset of FS( P ) . It is obvious to see that U := { U + K : K ∈ IS( P ) } is a clopenselector for FS( P ) . (cid:3) Note that we do not know if FS( P ) is canonically Skula whenever P is a well-quasi ordering:see Questions 6.2–6.6.Also let us remark that in special cases we can say more than in Proposition 4.8: Part (1)of the next result seems to be well-known, but we could not find it in the litterature. Proposition 4.11. (1) If P is an order-scattered linear ordering, then the space FS( P ) is a quotient of a successor ordinal, and thus FS( P ) has a tree-like canonical clopenselector. (2) If P is the disjoint union of two copies of ω then FS( P ) ∼ = ( ω + 1) is canonicallySkula but FS( P ) has no tree-like canonical clopen selector.Proof. (1) We set C = FS( P ) . So C is a complete chain, i.e. every subset of C has a supremumand an infimum, and C is a topological scattered space. Remark that the pointwise topologyon FS( P ) is the order topology on C .Since C is order-scattered, between any two elements of C there are two consecutive elementsand thus C is -dimensional. We prove the claim by induction of ht CB ( C ) . If ht CB ( C ) = 0 , C is finite and there is noting to prove. Next suppose that ht CB ( C ) = α . We assume that forevery complete and scattered chain D : if ht CB ( D ) < α then D is a continuous image of someordinal δ + 1 . Since C is -dimensional, it suffices to prove the result whenever C is unitary.We set c = min( C ) and c = max( C ) . The point e = endpt( C ) is called two-sided if [ c , e ) has no maximum and ( e, c ] has no minimum. We claim that we may assume that e is not two sided. Indeed, otherwise we split e , that is we replace e by two consecutiveelements e − < e + . So we obtain a chain b C = [ c , e − ] + [ e + , c ] satisfying ht CB ( b C ) = ht CB ( C ) and ∅ 6 = Endpt( b C ) ⊆ { e − , e + } . The identification of e − with e + defines an increasing andcontinuous map from b C onto C . Hence it suffices to prove the result whenever C := [ c , e − ] and endpt( C ) = e − . So max( C ) = e − := endpt( C ) . The case C := [ e + , c ] is similar.Let h c α i α<λ be a strictly increasing and unbounded sequence in [ c , e − ) . Since C is complete,we may assume that c = c and that sup β<α c β = c α for every limit α < λ . For each limit α < λ we add an immediate successor d α to c α that is: d α C and for every x ∈ C : if x > c α then x > d α . Hence we obtain a chain C = C ∪ { d α : α < λ } . The identification of d α with c α for all α , defines an increasing and continuous map from C onto C . So it suffices to prove theresult for C and thus, we may assume that C = C .We set C = [ c , c ] ( c := min( C ) ), and for each successor α ≥ let C α = [ d α , c α +1 ] . Since C = [ c , d ) and C α = ( c α , d α +1 ) , each C α is a clopen subset of C . Also for every limit α we set C α = { c α } (recall that c α has a successor d α in C ). So C is the lexicographic sum (cid:0)P α<λ C α (cid:1) + { max( C ) } . Now since ht CB ( C α ) < ht CB ( C ) , by the induction hypothesis, thereis a successor ordinal δ α and a continuous function f α from δ α onto C α . If α is limit, andthus C α = { c α } , we may assume that δ α = 1 . Moreover we set f λ (max( C )) = max( C ) .Hence f := S α ≤ λ f α is a continuous map from (cid:0)P α<λ δ α (cid:1) + { max( C ) } onto the well-ordering C := (cid:0)P α<λ C α (cid:1) + { max( C ) } .(2) Let P = ω ⊔ ω be the disjoint union of two copies of ω , that is P := ω ×{ } ∪ ω ×{ } and x and y are incomparable for any x ∈ ω ×{ } and y ∈ ω ×{ } . So IS( P ) ∼ = IS( ω ) × IS( ω ) ∼ = ( ω + 1) . By Telgàsky Theorem 3.3, the product of two unitary canonical Skulaspaces is canonically Skula. So ( ω + 1) is a unitary canonically Skula space.Now, by contradiction, assume that ( ω + 1) has a tree-like canonical clopen selector.By Proposition 4.6, ( ω + 1) is a quotient of α + 1 for some ordinal α and ( ω + 1) isretractable. But it is obvious that ( ω + 1) is not retractable: consider the closed subset ( ω +1) ×{ ω } ∪ { ω }× ( ω +1) of ( ω +1) . A contradiction. (cid:3) Mrówka spaces. Recall that a Mrówka space K A is a unitary canonical Skula space ofheight 2. The space K A is defined by an infinite almost disjoint family A on an infinite set S . We may assume that endpt( K A ) = max( K A ) (1) Let A be an infinite almost disjoint family on S . Then the space H ( K A ) is a unitarycanonical Skula space of height ω (reformulation of Theorems 4.1 and 4.4(3) –with α = 2 –. Therefore H ( K A ) is far from being a Mrówka space.(2) Let A be maximal almost disjont family on ω . Then the Mrówka space K A is nothomeomorphic to a topological semilattice (Proposition 5.3).We describe, in two ways, a general procedure of modifying an almost disjoint family A ona set S leading to a Mrówka space K A ⋆ with a continuous join operation. Recall that [ I ] <ω ∗ denotes the nonempty and finite subsets of I .On one hand, given A, B ∈ A with A = B , notice that [ A ] <ω ∩ [ B ] <ω = [ A ∩ B ] <ω is finite.Setting A ⋆ := [ A ] <ω ∗ , it follows that the family A ⋆ = { A ⋆ : A ∈ A } is almost disjoint on S ⋆ := [ S ] <ω . Therefore K A ⋆ := S ⋆ ∪ A ⋆ ∪ {∞ ⋆ } is a Mrówka space, where ∞ ⋆ = Max( K A ⋆ ) .On the other hand, we can describe K A ⋆ is a more formal way as follows. Since K A iscanonically Skula, by Theorem 4, H ( K A ) is a unitary canonical Skula space. We may assumethat ∞ ⋆ := endpt( H ( K A )) = max( H ( K A )) . Any member L := ↓ L of H ( K A ) with L = K A is of the form L = S(cid:8) A ∪ { x A } : A ∈ A L (cid:9) ∪ ρ L where A L & A is finite, ρ L is a finite subset IETORIS HYPERSPACES OVER PRIESTLEY SPACES . . . (July 15th, 2020) 27 of S and | A L | + | ρ L | ≥ . We set E = (cid:8) L ∈ H ( K A ) : | A L | ≥ , or | A L | = 1 | and | ρ L | ≥ (cid:9) . Obviously E is a closed final subset of h H ( K A ) , ⊆i : indeed H ( K A ) \ E = S { K + : | A K | ≤ } is an open initial subset of H ( K A ) . Hence the set E induces the closed equivalence relation E = { ( x, y ) ∈ H ( K A ) × H ( K A ) : x = y } ∪ ( E × E ) on H ( K A ) . Let G ( K A ) = H ( K A ) / E . Since we collapse only all elements of E in a point, denoted by ∞ , we have: G ( K A ) = (cid:8) L ∈ H ( K A ) : L ∈ [ S ] <ω ∗ or L ∈ A (cid:9) ∪ {∞} . We denote by q : H ( K A ) → G ( K A ) the quotient map. Obviously G ( K A ) is compact. Foreach L ∈ G ( K A ) \ {∞} , ht CB ( L ) ≤ , and thus G ( K A ) is of height 2 and unitary. So G ( K A ) is a Mrówka space. Moreover q ( L ) = L for any L ∈ K A ⋆ ) \ {∞ ⋆ } , and q ( ∞ ⋆ ) = ∞ := E . So,identifying ∞ ⋆ with ∞ ,( ⋆ ) The identity map Id : K A ⋆ → G ( K A ) (with ∞ ⋆ 7→ ∞ ) is a homeomorhism onto.To show that K A ⋆ := G ( K A ) has a structure of a continuous join operation ∨ , we need thefollowing fact that can be found in [19, Theorem 1.54]. Claim. Let h Y, m Y i be a compact topological join semilattice and let E be a closed nonemptyfinal subset of Y . Then the quotient space X := Y /E obtained by identification of all points of E has a continuous join operation m X . (cid:4) Since H ( K A ) is compact and a -dimensional join semilattice, by the claim, G ( K A ) has acontinuous join semilattice operation. We have proved the following result. Theorem 4.12. Let K A be a Mrówka pospace. (1) K A ⋆ = G ( K A ) and G ( K A ) is a Mrówka space with a continuous join operation and G ( K A ) has a canonical selector. (2) η : K A → K A ⋆ defined by η ( x ) = ↓ x for x ∈ K A is a one-to-one, increasing andcontinuous function. (cid:3) Now we will apply the above results to some examples of Mrówka pospace.4.3. Lusin families and ladder systems. An uncountable almost disjoint family A ofinfinite subsets of N is called a Lusin family (called inseparable family by Abraham and Shelahin [6]) if S A = N and for any subset H ⊆ N one of the families { A ∈ A : A ⊆ ∗ H } or { A ∈ A : A ⊆ ∗ X \ H } is countable. Here we denote by ⊆ ∗ the almost inclusion: for two sets A, B we write A ⊆ ∗ B if A \ B is finite. The first example of a Lusin family was constructed by Lusin [32] who actually constructed a “special Lusin family”. For completeness we give theproof of Proposition 4.13 (cf. [31, Ch. 3, Theorem 4.1]). Proposition 4.13 (Lusin) . There exists a Lusin family L on N of cardinality ℵ .Proof. We construct by transfinite induction pairwise almost disjoint sets A α ∈ [ N ] ω so thatfor each α < ω the following condition is satisfied:(L1) (cid:0) ∀ k ∈ ω (cid:1) (cid:0) | { ξ < α : A ξ ∩ A α ⊆ k } | < ℵ (cid:1) . We start by choosing arbitrary disjoint infinite sets A , A , . . . ⊆ N . Fix ω ≤ β < ω and sup-pose A ξ have been constructed for ξ < β . Enumerate { A ξ : ξ < β } as h B n : n ∈ ω i . Construct A β in such a way that(L2) (cid:0) ∀ n ∈ ω (cid:1) (cid:0) | A β ∩ B n \ ( B ∪ · · · ∪ B n − ) | = n (cid:1) . It is clear that (L1) holds. Thus, the construction can be carried out.We claim that L = h A α i α<ω is a Lusin family. By contradiction, suppose that H ⊆ N issuch that both sets L = { α < ω : A α ⊆ ∗ H } and R = { β < ω : A β ⊆ ∗ N \ H } are uncountable. So H is infinite. Refining L and R , we may assume that for some k ∈ ω theinclusion(L3) A α \ H ⊆ k and A β ∩ H ⊆ k holds for every α ∈ L and β ∈ R .Choose β ∈ R so that the set L ∩ β is infinite. Then, by (L1), for each k we can find ξ ( k ) ∈ L ∩ β such that A ξ ( k ) ∩ A β k . Choose x k ∈ A ξ ( k ) ∩ A β \ k . Then, by (L3), we concludethat x k ∈ H and therefore, since x k ∈ A β , we have x k ∈ A β ∩ H . Hence A β ∩ H is infinite,contradicting (L3). (cid:3) Comment. The crucial property of the almost disjoint family invented by Lusin is Condition(L1). A family A of infinite subsets of a countable set N is called a special Lusin family if itsatisfies condition (L1) with the quantifier “ ( ∀ n ∈ N ) ” replaced by “ ( ∀ s ∈ [ N ] <ω ) ”. That is:for each α < ω :( ⋆ ) ( ∀ s ∈ [ N ] <ω ) |{ ξ < α : A ξ ∩ A α ⊆ s }| < ℵ . Obviously, this property depends on the enumeration of the family.Therefore the above proof shows the existence of a special Lusin family. Also Lusin’s The-orem says that a special Lusin family is a Lusin family. (cid:4) The next results follows from Theorem 4.12. Proposition 4.14. Let L be a special Lusin family on N . IETORIS HYPERSPACES OVER PRIESTLEY SPACES . . . (July 15th, 2020) 29 (1) L ⋆ is a special Lusin family. (2) G ( K L ) = K L ⋆ and K L ⋆ admits a continuous join semilattice structure. (cid:3) If L ⊆ ω is a stationary set, then a ladder (system) over L is a sequence L = h c α i α<ω ( α ∈ L and α is a limit ordinal) such that each c α := h c α,n i n<ω is a strictly increasing ω -sequence cofinal in α . So L is an almost disjoint family on ω . We shall develop the laddersystem in a similar way as Lusin sequences. Proposition 4.15. Let L = { c δ : δ ∈ L } be a ladder system, where L denotes the set of allinfinite countable limit ordinals. Then there is a ladder system L ⋆ such that (1) L ⋆ has a structure of continuous join-semilattice. (2) There are a subset B of ω and a bijection h : ω → B such that L ⋆ = { h [ A ] : A ∈ L } .Proof. We set B δ = [ δ ] <ω and let B = S δ ∈ L B δ = [ ω ] <ω . Let C δ = [ c δ ] <ω . By definition, L ⋆ = { C δ : δ ∈ L } . We must show that L ⋆ is a ladder system. Define inductively a well-ordering on B , observing the following rule:Given α, β ∈ L such that β is the successor of α , the set B β \ B α has order type ω and B α is an initial segment of B β .This is clearly possible, because B β \ B α is infinite and countable whenever α < β . Finally,the ordering on B is isomorphic to ω and each C δ has order type ω , because C δ ∩ B α is finitewhenever α < δ . Thus, L ⋆ := { C δ } δ ∈ L is a ladder system. Now, by the construction, L ⋆ satisfies (1) and (2). (cid:3) Note that the above proof can be easily adapted to more general ladder systems, over astationary subset S of ω . The family L ⋆ appearing in Proposition 4.15, is an almost disjointfamily on ω and we may assume that ω = S L ⋆ . Therefore: Corollary 4.16. There exists a ladder system L ⋆ such that K L ⋆ is Mrówka space with acontinuous join operation. (cid:3) Complements on Hyperspaces and on Skula spaces Given a Priestley space, we complete the relationship between its hyperspace and its Vietorishyperspace, and we analyse the relationship between Skula spaces and topological semilattice.5.1. Priestley hyperspaces versus Vietoris hyperspaces. Let h X, ≤ X i and h Y, ≤ Y i betwo Priestley spaces and let f : X → Y be a continuous and increasing map. We consider themaps: η X : X → H ( X ) where η X ( x ) := ↓ x and η Y : Y → H ( Y ) where η Y ( y ) = ↓ y . Since f and η Y are increasing and continuous, ˆ η := η Y ◦ f is increasing and continuous. Soby Proposition 2.4, there exists a unique continuous join-semilattice homomorphism H ( f ) : H ( X ) → H ( Y ) such that H ( f ) ◦ η X = ˆ η where H ( X ) and H ( Y ) are endowed with the Priestleytopology T X and T Y respectively. So the following diagram( ⋆ ) X f (cid:15) (cid:15) ˆ η ' ' ◆◆◆◆◆◆◆◆◆◆◆◆◆ η X / / H ( X ) H ( f ) (cid:15) (cid:15) ✤✤✤ Y η Y / / H ( Y ) is commutative, and thus H ( f ) ◦ η X = ˆ η = η Y ◦ f .To a Priestley space X , we associate the same space X ′ with the equality relation. So X ′ isalso Priestley and H ( X ′ ) , denoted by exp( X ) , is the Vietoris hyperspace. Since the inclusionmap ı : X ′ → X is increasing and onto, ı defines an onto continuous semilattice homomorphism : H ( X ′ ) → H ( X ) satisfying ◦ η X ′ = ˆ η = η X ◦ ı . Note that is onto and thus H ( X ) is a continuous image of the compact space exp( X ) .On the other hand, considered as sets, we have, by the definition: H ( X ) ⊆ exp( X ) . We denote by T the topology on H ( X ) , by T ′ the topology on exp( X ) , and by T i := T ′ ↾ H ( X ) the induced topology T ′ of exp( X ) on H ( X ) . Proposition 5.1. With the above notation, T ⊆ T i and the following properties are equivalent: (i) H ( X ) is a closed subset of exp( X ) . (ii) T = T i .Proof. The inclusion map Id : h H ( X ) , T i i → h H ( X ) , T i is continuous. Indeed let U + be aclopen neighborhood of U in h H ( X ) , T i . So U + ∈ T where U is a clopen initial subset of X .Since U ∈ exp( X ) we have { K ∈ exp( X ) : K ⊆ U } ∩ H ( X ) = Id − [ U + ] = U + ∈ T i .(i) ⇒ (ii) Suppose that H ( X ) is closed in exp( X ) . Since Id is continuous, by the compactnessof h H ( X ) , T i i , we have T ′ ↾ H ( X ) := T i = T .(ii) ⇒ (i) Suppose T i = T . Since Id is continuous, by the compactness of h H ( X ) , T i , the set H ( X ) is closed in h exp( X ) , T ′ i . (cid:3) We apply the above result to show that for any canonically Skula space X , the topologies T and T i are distinct. Proposition 5.2. Let X be an infinite Skula space. Then H ( X ) is not closed in exp( X ) .Therefore, by Proposition , the topology on H ( X ) is not the induced topology on exp( X ) . IETORIS HYPERSPACES OVER PRIESTLEY SPACES . . . (July 15th, 2020) 31 Proof. Recall that [ I ] <ω ∗ denotes the set of nonempty finite subsets of I .Consider e ∈ X such that ht CB ( x ) = 1 . We may assume that there are a clopen selector U for X , e ∈ X and U e ∈ U such that its Cantor-Bendixson derivative U e [1] is { e } . By thedefinition, the space U e is homeomorphic to the Aleksandrov (one-point) compactification ofthe infinite discrete space Z := U e \ { e } . Since { e } is compact and e is not isolated in X , andthus { e } is not clopen in X , we have(1) { e } ∈ exp( X ) \ H ( X ) .Since X is canonically Skula, for x, y ∈ U e , we have x < y iff x ∈ Z and y = e . Hence H ( U e ) = [ Z ] <ω ∗ ∪ { U e } . Therefore U e is the unique non-isolated point of H ( U e ) . A base ofclopen neighborhoods of U e ∈ H ( X ) is the set of W σ := H ( U e ) \ [ σ ] <ω ∗ where σ ∈ [ Z ] <ω ∗ . Foreach σ ∈ [ Z ] <ω ∗ choose z σ ∈ Z \ σ . Then { z σ } 6∈ W σ and thus { e } is an accumulation point of (cid:8) { z σ } : σ ∈ [ Z ] <ω ∗ (cid:9) ⊆ H ( X ) in the space exp( X ) . We have proved that:(2) { e } is an accumulation point of H ( X ) in exp( X ) .Properties (1) and (2) show that H ( X ) is not closed in exp( X ) . (cid:3) Skula spaces and compact semilattices. In the rest of this section, we show that fora compact scattered space X , the following properties are independent.(1) X has a continuous semilattice operation.(2) X is a Skula space. Proposition 5.3 (Banakh and all [8]) . Let A be maximal almost disjont family on ω . Thenthe Mrówka space K A is not homeomorphic to any topological semilattice.Moreover K A is a separable canonical Skula space.Proof. The space K A is unitary and ht CB ( K A ) = 2 = rk WF ( K A ) . The choice of A as amaximal almost disjoint family (that is A is not contained in a strictly larger almost disjointfamily on ω ) guarantees that K A contains no sequence of isolated point that tend to ∞ . Thenby Theorem 3 of [8] the space K A cannot be homeomorphic to a topological semilattice. (cid:3) Recall that ( ω + 1) is the space of the FS( P ) where P := ω ⊔ ω is the disjoint union oftwo copies of ω .In the next result, the “non Skula” part was proved in terms of Boolean algebras by Bonnetand Rubin: Theorem 4.1 of [3]. For completeness we show this result using a shorter topologicalproof. Proposition 5.4. Let X be the quotient space of Y := ( ω + 1) by the closed “lower triangle” △ := {h β, γ i ∈ FS( P ) : β ≥ γ } . That is, X is the quotient space Y / ∼ where ∼ is the equiva-lence relation: x ∼ y if x, y ∈ △ or x = y . Then Y is canonically Skula, but X is not Skulaand X has a continuous semilatice operation. Boolean sketch. The Boolean algebra B of clopen subsets of Y := ( ω + 1) is generated by theset G := { ( α, β ] × ( γ, δ ] : α, β, γ, δ ≤ ω } . Let B ∗ be the Boolean subalgebra of B generatedby the set G ∗ := { ( α, β ] × ( γ, δ ] ∈ G : β ≤ γ } . Then B := Clop( Y ) and B ∗ := Clop( X ) arethe algebras appearing in Theorem 4.1 of [3]: B is canonically well-generated and B ∗ is nota well-generated subalgebra of B . In others words, Y has a canonical clopen selector, X is atopological quotient of Y but X has no clopen selector. (cid:3) Topological proof. Since Y is compact and △ is closed, X is Hausdorff and compact. We denoteby q : Y → X the quotient map.For a contradiction assume that X has a clopen selector U = { U x : x ∈ X } . For each y ∈ Y we set V y = q − [ U q ( y ) ] . Note that V y is a clopen neighborhood of y in X . For simplicity, V y = q − [ U q ( y ) ] is also denoted by V x where x := q ( y ) ∈ X .Since △ ∈ X := Y / ∼ and q ( t ) = △ for t ∈ △ ⊆ Y , the set V △ := q − [ U t ] is clopen in Y and V △ contains the triangle △ . So for every limit ordinal λ ≤ ω the set V △ is a neighborhood of h λ, λ i in Y and we can find an ordinal f ( λ ) < λ such that [ f ( λ ) , λ ] ⊆ V △ . By Fodor Theorem,there are a stationary set S ⊆ ω and γ ∈ ω such that f ↾ S = γ . We may assume that γ = min( S ) . Hence( ∗ ) [ γ, ω ) = S λ ∈ S [ γ, λ ] ⊆ V △ and [ γ, ω ] = cl Y ([ γ, ω ) ) ⊆ cl Y ( V △ ) = V △ where cl Y ( . ) denote the topological closure operation (in Y ).For every α < ω , since V h α,ω i := q − [ U q h α,ω i ] is a clopen neighborhood of h α, ω i in X , wecan find a countable ordinal g ( α ) ≥ α such that h α, g ( α ) i ∈ V h α,ω i . Take any point α ∈ S with α ≥ γ and by induction for every n ∈ ω choose an ordinal α n +1 ∈ S such that α n +1 > max( { α k : k ≤ n } ∪ { g ( α k ) : k ≤ n } ) and choose g ( α n +1 ) < ω such that h α n +1 , g ( α n +1 ) i ∈ V h α n +1 ,ω i . Let α ω = sup n ∈ ω α n = lim n ∈ ω α n .We claim that h α ω , α ω i ∈ V h α ω ,ω i . Since the set V h α ω ,ω i is closed, it suffices to check thateach clopen neighborhood W of h α ω , α ω i in X meets the set V h α ω ,ω i . From the fact that thesequence h α n i n ∈ ω converges to α ω , α ω = sup n g ( α n ) = lim n g ( α n ) , and that W and V h α ω ,ω i ,there is m ∈ ω such that [ α m , α ω ] ⊆ W , h α m , ω i ∈ V h α ω ,ω i and, by the choice of any g ( α m ) , h α m , g ( α m ) i ∈ V h α m ,ω i . Since h α m , ω i ∈ V h α ω ,ω i and since U is a clopen selectorfor X , V h α m ,ω i ⊆ V h α ω ,ω i and thus h α m , g ( α m i ∈ V h α ω ,ω i . On the other hand, h α n , g ( α n ) i ∈ [ α n , α ω ] ⊆ W . Thus W ∩ U h α ω ,ω i is nonempty and hence h α ω , α ω i ∈ V h α ω ,ω i .Now since h α ω , α ω i ∈ V h α ω ,ω i , by the definition of Y / △ := Y / ∼ , we have q ( △ ) = q ( h α ω , α ω i ) ∈ U h α ω ,ω i . The fact that U is a clopen selector implies that U △ ⊆ U h α ω ,ω i . On the other hand, h α ω , ω i ∈ [ γ, ω ] ⊆ V △ and thus q ( h α ω , ω i ) ∈ U q ( △ ) . Again since U is a clopen selec-tor, U h α ω ,ω i ⊆ U q ( △ ) . Therefore U q ( h α ω ,ω i ) = U q ( △ ) and thus q ( h α ω , ω i ) = q ( △ ) := △ . But q ( h α ω , ω i ) 6∈ △ because α ω = ω . This contradiction shows that X := Y / ∼ is not a Skulaspace. IETORIS HYPERSPACES OVER PRIESTLEY SPACES . . . (July 15th, 2020) 33 Next the continuous join operation ∗ : Y × Y → Y defined by ( β, γ ) ∗ ( β ′ , γ ′ ) = (min { β, β ′ } , max { γ, γ ′ } ) induces a continuous semilattice operation ∨ on X := Y / ∼ , defined by ( u/ ∼ ) ∨ ( v/ ∼ ) :=( u ∗ v ) / ∼ for any u, v ∈ Y because the singletons and △ are closed: see also [19, Theorem1.54]. (cid:3) Final remarks and open questions Recall that any countable scattered compact space is homeomorphic to a countable successorordinal. In Proposition 5.4, we have seen that there is a canonical Skula space with a nonSkula quotient space. By “duality” we ask the following question. Question 6.1. Is there an uncountable compact space such that every closed subspace iscanonically Skula? (cid:4) In §4.1, we have seen that for a poset P , the space h FS( P ) , ⊇i of all final subsets of P and space h IS( P ) , ⊆i of all initial subsets of P endowed with the pointwise topology T p areorder-isomorphic and homeomorphic. So( ⋆ ) We identify the Priestley spaces FS( P ) and IS( P ) . In Proposition 4.10, we have seen that if P is a well-quasi ordering (well-founded and anyset of pairwise incomparable elements is finite), then FS( P ) is a Skula space. From this result,M. Pouzet asks for the following question. Question 6.2 (M. Pouzet) . Let P be a well-quasi ordering. Is FS( P ) canonically Skula? (cid:4) Question 6.3. Let P be a narrow order-scattered poset (and thus FS( P ) is Skula). Is FS( P ) canonically Skula? (cid:4) In [3, Theorem 2.1], Bonnet and Rubin proved that every quotient space of ( ω +1) × ( ω +1) iscanonically Skula, and in Proposition 5.4 we have seen that ( ω + 1) has a non-Skula quotientspace. These facts, in a “dual” way, ask for the following question. Question 6.4. (1) Is every closed subset of ( ω + 1) × ( ω + 1) canonically Skula?(2) Is every closed subset of ( ω + 1) canonically Skula? (cid:4) A partial ordering h P, ≤i has finite width , if for some n ∈ ω , P is the union of n chains.Note that by Dilworth Theorem, a poset P has finite width whenever there is n ∈ ω such thatevery antichain of P has cardinality ≤ n . Questions 6.2–6.4 ask also for the following. Question 6.5. Let P be a well-founded poset of finite width. Is FS( P ) canonically Skula? (cid:4) So we ask for similar cases. Question 6.6. (1) Let P be a well-founded poset of finite width. Is every closed subsetof FS( P ) canonically Skula? (2) More generally let P be a narrow and order-scattered poset. Is every closed subset of FS( P ) canonically Skula? (cid:4) In view of Proposition 5.2 we ask for the following question. Question 6.7. Characterize the non trivial Priestley spaces X such that H ( X ) is closed in exp( X ) . (cid:4) References [1] U. Abraham, R. Bonnet: Every Superatomic Subalgebra of an Interval Algebra is Embeddable in anOrdinal Algebra , Proc. of the Amer. Math. 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