aa r X i v : . [ m a t h . L O ] O c t DESCRIPTIVE PROPERTIES OF HIGHER KUREPA TREES
PHILIPP L ¨UCKE AND PHILIPP SCHLICHT
Abstract.
We use generalizations of concepts from descriptive set theoryto study combinatorial objects of uncountable regular cardinality, focussingon higher Kurepa trees and the representation of the sets of cofinal branchesthrough such trees as continuous images of function spaces. For different typesof uncountable regular cardinals κ , our results provide a complete pictureof all consistent scenarios for the representation of sets of cofinal branchesthrough κ -Kurepa trees as retracts of the generalized Baire space κ κ of κ . Inaddition, these results can be used to determine the consistency of most of thecorresponding statements for continuous images of κ κ . Introduction
Given an infinite regular cardinal κ , the generalized Baire space of κ consistsof the set κ κ of all functions from κ to κ equipped with the topology whose basicopen sets are of the form N s = { x ∈ κ κ | s ⊆ x } , where s is an element of the set <κ κ of all functions t : α −→ κ with α < κ . One of the most basic structuralfeatures of the classical Baire space ω ω is the fact that non-empty closed subsetsof ω ω are retracts of ω ω (see [7, Proposition 2.8]), i.e. for every such that C thereis a continuous surjection r : ω ω −→ A with r ↾ A = id A . In contrast, the resultsof [10] show that such results cannot be generalized to higher cardinalities and thisfailure highlights fundamental differences between ω and higher regular cardinals,e.g. the existence of limit ordinals below the given cardinal and possible existenceof Aronszajn and Kurepa trees at these cardinals. First, [10, Proposition 1.4] showsthat for every uncountable regular cardinal κ , there is a non-empty closed subsetof κ κ that is not a retract of κ κ . Moreover, [10, Theorem 1.5] shows that for everyuncountable cardinal κ satisfying κ = κ <κ , there is a non-empty closed subset of κ κ that is not a continuous image of κ κ . In particular, the classes of continuousimages of κ κ and continuous images of non-empty closed subsets of κ κ in κ κ donot coincide in this case. Finally, the various results of [10] strongly motivate aninvestigation of the class of all closed subsets of κ κ that are continuous images of κ κ . Since closed subsets of κ κ canonically correspond to sets of cofinal branchesthrough trees of height κ , the study of this class of subsets turns out to be closely Mathematics Subject Classification.
Key words and phrases.
Generalized Baire spaces, Kurepa trees, Continuous images.This project has received funding from the European Union’s Horizon 2020 research and in-novation programme under the Marie Sk lodowska-Curie grant agreements No 842082 of the firstauthor (Project
SAIFIA: Strong Axioms of Infinity – Frameworks, Interactions and Applications )and No 794020 of the second author (Project
IMIC: Inner models and infinite computations ). Thesecond author was partially supported by FWF grant number Y1012. It is easy to see that, if there exists a cardinal µ < κ with 2 µ = 2 κ , then every non-emptysubset of κ κ is a continuous image of κ κ . In particular, some cardinal arithmetic assumption on κ is necessary for the conclusion of [10, Theorem 1.5]. connected to the existence of certain combinatorial objects of the given cardinalityand the validity of combinatorial principles at the corresponding cardinals.In this paper, motivated by results from [10] (see Theorem 2.1 below), we focuson the representation of the sets of cofinal branches through κ -Kurepa trees ascontinuous images of κ κ . Our results will show that if κ is a successors of a cardinalof countable cofinality, then κ -Kurepa trees provide more examples of closed subsetsof κ κ that are not continuous images of κ κ . In contrast, we will also show that forother types of uncountable regular cardinals, various natural questions about suchcontinuous representations of sets of branches through Kurepa trees can have widelydifferent answers that depend heavily on the underlying model of set theory.Remember that a partial order T is a tree if it has a unique minimal element andfor every element t of T , the set pred T ( t ) = { s ∈ T | s < T t } is well-ordered by < T .Given a tree T , t ∈ T and α ∈ Ord, we define lh T ( t ) = otp (pred T ( t ) , < T ), T α = { s ∈ T | lh T ( s ) = α } , T <α = S { T ¯ α | ¯ α < α } and ht( T ) = min { β ∈ Ord | T β = ∅} .Next, we say that a subset b of a tree T is branch if it < T -downwards-closed andlinearly ordered by < T and it is a cofinal branch if in addition otp ( b, < T ) = ht( T )holds. Finally, given an uncountable regular cardinal κ , a tree T with ht( T ) = κ isa κ -Kurepa tree if | T α | < κ holds for all α < κ and T has at least κ + -many cofinalbranches. We usually say Kurepa tree instead of ω -Kurepa tree. Seminal resultsof Jensen (see [2, Chapter III, Section 3]) show that, in the constructible universeL, κ -Kurepa trees exist for every uncountable regular cardinal κ . In contrast, aclassical argument of Silver (see [5, Section 3]) shows that if κ is an uncountableregular cardinal that is not inaccessible, θ > κ is inaccessible and G is Col( κ, <θ )-generic over V, then there are no κ -Kurepa trees in V[ G ].Fix an infinite regular cardinal κ and 0 < n < ω . A subset T of ( <κ κ ) n is a subtree of ( <κ κ ) n if dom( t ) = . . . = dom( t n − ) and h t ↾ α, . . . , t n − ↾ α i ∈ T forall h t , . . . , t n − i ∈ T and α < dom( t ). Note that for every such subtree T , thepartial order T T = h T, ⊳ i with h s , . . . , s n − i ⊳ h t , . . . , t n − i ⇐⇒ s ( t ∧ . . . ∧ s n − ( t n − for all h s , . . . , s n − i , h t , . . . , t n − i ∈ T is a tree with ( T T ) α = T ∩ ( α κ ) n for all α < κ . Set ht( T ) = ht( T T ) ≤ κ . If ht( T ) = κ , then there is a canonical bijectionbetween the set of cofinal branches through T T and the set[ T ] = {h x , . . . , x n − i ∈ ( κ κ ) n | ∀ α < κ h x ↾ α, . . . , x n − ↾ α i ∈ T } given by restrictions. It is easy to see that for every subtree T of ( <κ κ ) n of height κ , the set [ T ] is closed in the product space ( κ κ ) n . Moreover, for every non-emptyclosed subset C of ( κ κ ) n , the set T C = {h t , . . . , t n − i ∈ ( <κ κ ) n | C ∩ ( N t × . . . × N t n − ) = ∅} is a subtree of ( <κ κ ) n with C = [ T C ].Now, let κ be an uncountable regular cardinal and let T be a tree of height κ . If T is extensional at limit levels (i.e. if s = t holds for all s, t ∈ T withlh T ( s ) = lh T ( t ) ∈ Lim and pred T ( s ) = pred T ( t )), then it is easy to constructa subtree T of <κ κ with the property that the tree T and T T are isomorphic. Ingeneral, we can consider the tree ¯ T that consists of all non-empty branches b through Note that for all inaccessible cardinals κ , the existence of κ -Kurepa trees is trivial. Therefore,one usually considers trees with stronger restrictions on the size of levels at inaccessible cardinals(see [6] and Section 1.1 below). ESCRIPTIVE PROPERTIES OF HIGHER KUREPA TREES 3 T with the property that there exists t ∈ T with s ≤ T t for all s ∈ b and is orderedby inclusion. Then ¯ T is extensional at limit levels, the map [ t pred T ( t ) ∪ { t } ] isan isomorphism between T and a cofinal subtree of ¯ T and hence there is a canonicalisomorphism between the sets of cofinal branches through T and ¯ T . Moreover, itis easy to see that the assumption that T is a κ -Kurepa tree implies that ¯ T is a κ -Kurepa tree too. In combination, this shows that the existence of a κ -Kurepatree is equivalent to the existence of a κ -Kurepa subtree of <κ κ , i.e. a subtree T of <κ κ with the property that the tree T T is a κ -Kurepa tree.1.1. Kurepa trees that are not continuous images.
The following result pro-vides several scenarios in which the closed sets induced by Kurepa trees are notcontinuous images of the corresponding generalized Baire space. The statementof Corollary 1.2 below was our original motivation for the work presented in thispaper.
Theorem 1.1. If κ is an uncountable regular cardinal with µ ω ≥ κ for some µ < κ and T is a κ -Kurepa subtree of <κ κ with | [ T ] | > κ <κ , then the set [ T ] is not acontinuous image of κ κ . This theorem has the following two direct corollaries:
Corollary 1.2.
Assume that CH holds. If T is a subtree of <ω ω that is a Kurepatree, then [ T ] is not a continuous image of ω ω . (cid:3) Corollary 1.3.
Assume that κ = µ + = 2 µ for some singular cardinal µ of countablecofinality. If T is a subtree of <κ κ that is a κ -Kurepa tree, then [ T ] is not acontinuous image of κ κ . (cid:3) Note that the above notion of Kurepa trees trivializes at inaccessible cardinals κ , because the complete binary tree <κ r : κ κ −→ κ r ( x )( α ) =min { , x ( α ) } for all x ∈ κ κ and α < κ is a retraction from κ κ to κ Theorem 1.4.
Let M be an inner model and let κ be a cardinal with κ = κ <κ .If κ is inaccessible in M and ( κ + ) M = κ + holds, then there is a κ -Kurepa subtree T of <κ κ with T ⊆ ( <κ M and the property that the set [ T ] is not a continuousimage of κ κ . Motivated by these results about inaccessible cardinals, we also consider thefollowing strengthening of the definition of Kurepa trees: given an uncountableregular cardinal κ , a tree T of height κ is called slim if | T ( α ) | ≤ | α | holds forco-boundedly many α < κ . Classical results of Jensen and Kunen in [6] show thatthere are no slim κ -Kurepa trees at ineffable cardinals κ and, in the constructibleuniverse L, slim κ -Kurepa trees exist at every uncountable regular cardinal κ thatis not ineffable.The proof of Theorem 1.1 also allows us to derive the following statement. Theorem 1.5. If κ is an inaccessible cardinal and T is a slim Kurepa subtree of <κ κ , then [ T ] is not a continuous image of κ κ . PHILIPP L¨UCKE AND PHILIPP SCHLICHT
Finally, the results of this paper allow us to use results of Donder from [3] andVelleman from [14] to show that the absence of large cardinals in the constructibleuniverse implies the existence of Kurepa trees whose induced closed subsets are notcontinuous images.
Theorem 1.6.
Let κ be an uncountable regular cardinal such that κ = κ <κ holdsand neither κ nor κ + are inaccessible in L . Then there is a κ -Kurepa subtree of κ κ with the property that [ T ] is not a continuous image of κ κ . Kurepa trees that are continuous images.
Somewhat surprisingly, Corol-laries 1.2 and 1.3 in previous section turn out to be the only provable restrictionson the existence of κ -Kurepa whose sets of cofinal branches are continuous imagesof κ κ . The proof of the following theorem relies on a result of Donder on the struc-tural properties of the Kurepa trees constructed from the canonical morasses atsuccessor cardinals in the constructible universe. This result can be combined withTheorem 1.1 to show that the statement that there is an ω -Kurepa subtree tree T of <ω ω with the property that [ T ] is a continuous image of ω ω is independent ofthe axioms of ZFC by considering the constructible universe L and models of thenegation of the Continuum Hypothesis. Theorem 1.7.
Assume that
V = L holds. Then the following statements areequivalent for every uncountable regular cardinal κ : (i) The cardinal κ is not the successor of a cardinal of countable cofinality. (ii) There is a κ -Kurepa subtree of <κ κ with the property that [ T ] is a retractof κ κ . (iii) There is a κ -Kurepa subtree of <κ κ with the property that [ T ] is a contin-uous image of κ κ . In combination with Theorem 1.5, the previous result directly implies an analo-gous characterization for slim Kurepa trees.
Corollary 1.8.
Assume that
V = L holds. Then the following statements areequivalent for every uncountable regular cardinal κ : (i) The cardinal κ is neither inaccessible nor the successor of a cardinal ofcountable cofinality. (ii) There is a slim κ -Kurepa subtree of <κ κ with the property that [ T ] is aretract of κ κ . (iii) There is a slim κ -Kurepa subtree of <κ κ with the property that [ T ] is acontinuous image of κ κ . (cid:3) The next result shows that the positive implications of Theorem 1.7 for successorsof regular cardinals can also be obtained by collapsing an inaccessible cardinal tobecome the successor of an uncountable regular cardinal.
Theorem 1.9. If κ is an inaccessible cardinal and µ < κ is an uncountable regularcardinal, then there is a generic extension V[ G ] of the ground model V that preservesall cofinalities less than or equal to µ and greater than or equal to κ , such that thefollowing statements hold in V[ G ] : (i) κ = µ + . (ii) There is a κ -Kurepa subtree T of <κ κ with the property that the set [ T ] is a retract of κ κ . ESCRIPTIVE PROPERTIES OF HIGHER KUREPA TREES 5 (iii)
There is a κ -Kurepa subtree T of <κ κ with T ⊆ T and the property thatthe set [ T ] is not a continuous image of κ κ . The notion of slimness considered in Theorem 1.5 has a natural weakening thatonly requires | T ( α ) | ≤ | α | to hold in a stationary set of α < κ . We refer to thisproperty as stationary slimness . The following result shows that, in general, itis not possible to replace slimness by stationary slimness in the assumptions ofTheorem 1.5. Remember that an inaccessible cardinal κ is a 2 -Mahlo cardinal ifthe set of Mahlo cardinals less than κ is stationary in κ . Theorem 1.10. If κ is a -Mahlo cardinal, then the following statements hold ina generic extension of the ground model V : (i) κ is inaccessible. (ii) There is a stationary slim κ -Kurepa subtree T of <κ κ with the propertythat [ T ] is a retract of κ κ . Kurepa trees that are not retracts.
The trees constructed in the proofs ofthe results of the previous section have many isolated points. The next result showsthat this is a necessary condition. Note that, throughout this paper, inaccessible means strongly inaccessible and, if κ is an inaccessible cardinal, then <κ κ -Kurepa subtree of <κ κ and the set [ <κ
2] = κ κ κ without isolatedpoints. Theorem 1.11.
Let κ be an uncountable regular cardinal and let T be a κ -Kurepasubtree of κ κ . Assume that either κ is not inaccessible or T is stationary slim. Ifthe set [ T ] is a retract of κ κ , then it contains isolated points. This result allows us to show that the existence of Kurepa trees implies theexistence of Kurepa trees that are not retracts.
Theorem 1.12.
Let κ be an uncountable regular cardinal with κ <κ = κ . If thereis a κ -Kurepa tree S , then there is a κ -Kurepa subtree T of <κ κ with the propertythat the set [ T ] is not a retract of κ κ . Moreover, if S is stationary slim, then T canbe taken to be stationary slim. Finally, our techniques allow us to show that for higher Kurepa trees, the prop-erty of being a continuous image neither implies the existence of isolated branchesnor the property of being a retract.
Theorem 1.13.
Let κ be an uncountable regular cardinal. If there is a κ -Kurepatree S with the property that the set [ S ] is a continuous image of κ κ , then there is a κ -Kurepa subtree T of <κ κ with the property that the set [ T ] does not contain isolatedpoints, it is a continuous image of κ κ and it is not a retract of κ κ . Moreover, if S is stationary slim, then T can be taken to be stationary slim. Wide subtrees
The following result from [10] will be our main tool for showing that the closedsubsets induced by certain Kurepa trees are not continuous images of the wholespace.
Theorem 2.1 ([10, Theorem 7.1]) . Let κ be an uncountable regular, let A be anunbounded subset of κ and let T be a subtree of <κ κ . If µ is a cardinal with µ <κ < PHILIPP L¨UCKE AND PHILIPP SCHLICHT | [ T ] | and c : κ µ −→ [ T ] is a continuous surjection, then there is a strictly increasingsequence h λ n ∈ A | n < ω i with least upper bound λ and an injection i : Y n<ω λ n −→ T ( λ ) . such that x ↾ n = y ↾ n ⇐⇒ i ( x ) ↾ λ n = i ( y ) ↾ λ n holds for all x, y ∈ Q n<ω λ n and all n < ω . The above lemma allows us to provide short proofs of two theorems presentedin Section 1.1.
Proof of Theorem 1.1.
Let κ be an uncountable regular cardinal with µ ω ≥ κ forsome µ < κ and let T be a κ -Kurepa subtree of <κ κ with | [ T ] | > κ <κ . Assume,towards a contradiction, that the set [ T ] is a continuous image of κ κ . Since wehave κ <κ < | [ T ] | , we can apply Theorem 2.1 to find a strictly increasing sequence h λ n | n < ω i of ordinals in the interval ( µ, κ ) with least upper bound λ and aninjection i : Q n<ω λ n −→ T ( λ ). Moreover, since λ n > ν holds for all n < ω , wecan conclude that | T ( λ ) | ≥ | Y n<ω λ n | ≥ µ ω ≥ κ, contradicting the fact that T is a κ -Kurepa subtree of <κ κ . (cid:3) Proof of Theorem 1.5.
Let κ be an inaccessible cardinal and let T be a slim κ -Kurepa subtree of <κ κ . Assume, towards a contradiction, that the set [ T ] is acontinuous image of κ κ . Pick α < κ with the property that | T ( β ) | = | β | holdsfor all α ≤ β < κ . Using Theorem 2.1, we find a strictly increasing sequence h λ n | n < ω i of cardinals in the interval ( α, κ ) with least upper bound λ and aninjection i : Q n<ω λ n −→ T ( λ ). Then K¨onig’s Theorem allows us to conclude that | T ( λ ) | ≥ | Y n<ω λ n | ≥ | Y n<ω λ n +1 | > | X n<ω λ n | = λ, a contradiction. (cid:3) In the remainder of this section, we prove Theorem 1.6. Our arguments will bebased on the concept introduced in the next definition.
Definition 2.2. ([12, Section 1]) A tree T has a Cantor subtree if there is a strictlyincreasing sequence h λ n | n < ω i with λ = sup n<ω λ n < ht( T ) and an uncountablesubset B of T ( λ ) with the property that the set { s ∈ T ( λ n ) | ∃ t ∈ B s < T t } iscountable for every n < ω .The next statement is a direct consequence of Theorem 2.1. Corollary 2.3.
Let κ be an uncountable regular cardinal and and let T be a subtreeof <κ κ with | [ T ] | > κ <κ . If [ T ] is a continuous image of κ κ , then T contains aCantor subtree. (cid:3) Proof of Theorem 1.6.
Let κ be an uncountable regular cardinal with κ = κ <κ andthe property that neither κ nor κ + are inaccessible in L. Claim.
There is a simplified ( κ, -morass with linear limits (see [14, Section 2] ). ESCRIPTIVE PROPERTIES OF HIGHER KUREPA TREES 7
Proof of the Claim.
As discussed at the end of [11], our assumption allows us tofind A ⊆ κ such that κ + = ( κ + ) L[ A ] , Donder’s construction of a simplified ( κ, A ] and the resulting morassis also a simplified ( κ, (cid:3) By the above claim, we can apply [14, Theorem 4.3] to find a κ -Kurepa subtree T of <κ κ without Cantor subtrees. By Corollary 2.3, we know that the closed set[ T ] is not a continuous image of κ κ . (cid:3) Trees induced by inner models
This section is devoted to the proof of Theorem 1.4. Our arguments are avariation of the proof of [10, Theorem 1.5]. In addition, we will show that thestatement of Corollary 2.3 can, in general, not be reversed.
Proof of Theorem 1.4.
In the following, we let ≺· , ·≻ : Ord × Ord −→ Ord denotethe
G¨odel pairing function . Given an ordinal γ closed under ≺· , ·≻ and x ∈ γ
2, wedefine < x to be the unique binary relation on γ with α < x β ⇐⇒ x ( ≺ α, β ≻ ) = 1for all α, β < γ .Let κ be a cardinal satisfying κ = κ <κ and let M be an inner model with theproperty that κ is inaccessible in M and ( κ + ) M = κ + holds. Work in M and set W = { x ∈ κ | ( κ, < x ) is a well-order } . Then it is easy to see that the fact that κ is uncountable and regular implies that W is a closed subset of κ κ and hence there is a subtree T of <κ W = [ T ]. Since | W | = 2 κ , T ⊆ <κ κ is inaccessible, we can conclude that T is a κ -Kurepasubtree of <κ κ .Now, work in V. Then our assumptions on M imply that T is still a κ -Kurepasubtree of <κ κ . Moreover, if x ∈ [ T ], then the regularity of κ implies that ( κ, < x )is a well-order. Given x ∈ [ T ] and α < κ , we let rnk x ( α ) denote the rank of α in the well-order ( κ, < x ). Assume, towards a contradiction, that the set [ T ] is acontinuous image of κ κ . Then a combination of [10, Lemma 2.2] with [10, Lemma2.3] yields a <κ -closed subtree U of <κ κ × <κ κ without end nodes such that[ T ] = p [ U ] = { x ∈ κ κ | ∃ y h x, y i ∈ [ U ] } . Note that these properties of U imply that for every h t, u i ∈ U , there exists a pair h x, y i ∈ [ U ] with t ⊆ x and u ⊆ y . Given h t, u i ∈ U and α < κ , we define r ( t, u, α ) = sup { rnk x ( α ) | h x, y i ∈ [ U ] , t ⊆ x, u ⊆ y } ≤ κ + . Then r ( ∅ , ∅ , α ) = κ + holds for all α < κ , because the assumptions ( κ + ) M = κ + implies that for every γ < κ + , there exists x ∈ [ T ] M ⊆ [ T ] = p [ U ] with rnk x ( α ) ≥ γ and hence there exists y ∈ κ κ such that the pair h x, y i witnesses that r ( ∅ , ∅ , α ) ≥ γ . Claim. If γ < κ + , ( t, u ) ∈ U and α < κ with r ( t, u, α ) = κ + , then there is ( v, w ) ∈ U and α < β < dom( v ) such that t ( v , u ( w , dom( v ) is closed under ≺· , ·≻ , β < v α and r ( v, w, β ) ≥ γ . A subtree S of ( <κ κ ) n is <κ -closed if for every λ < κ and every ⊳ -increasing sequence hh s ξ , . . . , s ξn − i | ξ < λ i in S , the tuple h S ξ<λ s ξ , . . . , S ξ<λ s ξn − i is also an element of S . We callan element of such a tree S an end node if it is ⊳ -maximal in S . PHILIPP L¨UCKE AND PHILIPP SCHLICHT
Proof of the Claim.
By our assumptions, we can find h x, y i ∈ U with t ⊆ x , u ⊆ y and rnk x ( α ) ≥ γ + κ . Then there is α < β < κ with β < x α and rnk x ( β ) ≥ γ .Pick ξ > β + dom( t ) closed under ≺· , ·≻ . Then t ⊆ x ↾ ξ , u ( y ↾ ξ , β < x ↾ ξ α and r ( x ↾ ξ, y ↾ ξ, β ) ≥ γ . (cid:3) Claim. If h t, u i ∈ U and α < κ with r ( t, u, α ) = κ + , then there exists ( v, w ) ∈ U and α < β < dom( v ) such that t ( v , u ( w , dom( v ) is closed under ≺· , ·≻ , β < v α and r ( v, w, β ) = κ + .Proof of the Claim. For each γ < κ + , let h v γ , w γ i ∈ U and α < β γ < dom( t γ ) bethe objects given by the above claim. Since κ = κ <κ holds, we can find h v, w i ∈ U , β < κ and X ⊆ κ + with | X | = κ + such that v γ = v , w γ = w and β γ = β for all γ ∈ X . But this implies that r ( v, w, β ) = κ + . (cid:3) Using the last claim, we now construct sequences hh t n , u n i ∈ U | n < ω i and h α n < κ | n < ω i such that dom( t n +1 ) is closed under ≺· , ·≻ , t n ( t n +1 , u n ( u n +1 , α n < α n +1 < dom( t n +1 ) and α n +1 < t n +1 α n for all n < ω . Set t = S n<ω t n and u = S n<ω u n . Then the properties of U imply that h t, u i ∈ U and there exists x ∈ [ T ] with t ⊆ x . But then ( κ, < x ) is a well-order with α n +1 < x α n for all n < ω ,a contradiction. (cid:3) As promised above, we end this section by showing that, for certain Kurepatrees, the converse of the statement of Corollary 2.3 does not hold true.
Corollary 3.1.
Let µ be an uncountable regular cardinal, let θ > κ be inaccessiblecardinals above µ , let G be Col( κ, <θ ) -generic over V and let H be Col( µ, <κ ) -generic over V[ G ] . Then the following statements hold in V[ G, H ] : (i) There is a κ -Kurepa subtree T of <κ κ with the property that the set [ T ] isnot a continuous image of κ κ . (ii) Every κ -Kurepa tree contains a Cantor subtree.Proof. First, since ( κ + ) V[ G,H ] = θ = ( κ + ) V[ G ] and κ is inaccessible in V[ G ], we canapply Theorem 1.4 to conclude that, in V[ G, H ], there is a κ -Kurepa subtree T of <κ κ such that the set [ T ] is not a continuous image of κ κ . Next, fix a Col( µ, <κ )-nice name ˙ T ∈ V[ G ] for a κ -Kurepa tree with underlying set κ . Since Col( µ, <κ )satisfies the κ -chain condition in V[ G ] and Col( κ, <θ ) satisfies the θ -chain conditionin V, we can find κ < ϑ < θ with the property that ˙ T ∈ V[ G ϑ ], where G ϑ = G ∩ Col( κ, <ϑ ). Then ˙ T H ∈ V[ G ϑ , H ], θ is inaccessible in V[ G ϑ , H ] and henceforcing with Col( κ, [ ϑ, θ )) V over V[ G ϑ , H ] adds a new cofinal branch to ˙ T H . Sincethe partial order Col( κ, [ ϑ, θ )) V is <µ -closed in V[ G ϑ , H ] and µ is an uncountablecardinal in V[ G ϑ , H ], standard arguments show that ˙ T H contains a Cantor subtreein V[ G ϑ , H ] and this subtree is still a subtree in V[ G, H ]. (cid:3) Superthin Kurepa trees
The concepts introduced in the next definition will play a central role in ourproofs of Theorems 1.7, 1.9 and 1.10.
Definition 4.1.
Let κ be an infinite regular cardinal and let T be a subtree of <κ κ . (i) The tree T is pruned if for every s ∈ T , there is t ∈ T with s ( t . ESCRIPTIVE PROPERTIES OF HIGHER KUREPA TREES 9 (ii) The boundary ∂T of T is defined as the set of minimal elements of <κ κ \ T ,i.e. ∂T = { t ∈ <κ κ \ T | ∀ α ∈ dom( t ) t ↾ α ∈ T } . (iii) The tree T is superthin if | ( T ∪ ∂T ) ∩ α κ | < κ holds for all α ∈ Lim ∩ κ .Note that a subtree T of <κ κ of height κ is <κ -closed if and only if ∂T ∩ α κ = ∅ holds for all α ∈ Lim ∩ κ . The following observation shows that the fact that allnon-empty closed subsets of ω ω are retracts of ω ω can be generalized to a certainclass of closed subsets of higher Baire spaces. Proposition 4.2. If κ is an infinite regular cardinal and T is a <κ -closed prunedsubtree of <κ κ , then N t ∩ [ T ] = ∅ for every t ∈ T and the closed set [ T ] is a retractof κ κ .Proof. Given t ∈ T , our assumptions on T allow us to do an easy inductive con-struction that produces a sequence h t α | α < κ i of elements of T such that t = t ,dom( t α ) = dom( t ) + α and t α ⊆ t β for all α ≤ β < κ . In this situation, we have x t = S { t α | α < κ } ∈ N t ∩ [ T ].Now, fix y ∈ κ κ . Then there is a unique β < κ with y ↾ β ∈ ∂T and, since T is <κ -closed, we know that β is not a limit ordinal. Hence there is a unique α y < κ with y ↾ α y ∈ T and y ↾ ( α y + 1) / ∈ T .Let r : κ κ −→ [ T ] denote the unique function with r ↾ [ T ] = id [ T ] and r ( y ) = x y ↾ α y for all y ∈ κ κ \ [ T ]. Claim.
The function r is continuous.Proof of the Claim. First, fix x ∈ [ T ], α < κ and y ∈ N x ↾ α . If y ∈ [ T ], then wehave r ( y ) = y ∈ N x ↾ α = N r ( x ) ↾ α . In the other case, if y / ∈ [ T ], then x ↾ α ∈ T implies that α y ≥ α and hence r ( y ) = x y ↾ α y ∈ N x ↾ α = N r ( x ) ↾ α .Now, fix x ∈ κ κ \ [ T ] and α < κ . Pick α ≤ β < κ with x ↾ β / ∈ T and y ∈ N x ↾ β .Then we have y / ∈ [ T ], α x = α y ≤ β , x ↾ α x = y ↾ α y and r ( y ) = x y ↾ α y = x x ↾ α x = r ( x ) ∈ N r ( x ) ↾ α . (cid:3) Since r ↾ [ T ] = id [ T ] , the above claim completes the proof of the proposition. (cid:3) Lemma 4.3.
Let κ be an uncountable regular cardinal. If there is a superthin κ -Kurepa subtree S of <κ κ , then there is a κ -Kurepa subtree T of <κ κ with theproperty that [ T ] is a retract of κ κ . Moreover, if S is pruned, then T can be takento contain S as a subtree.Proof. Let S be a superthin κ -Kurepa subtree of <κ κ . If S is pruned, then weset S = S . Otherwise, we define S = { t ∈ S | ∃ x ∈ [ T ] t ⊆ x } . Then it is easy to see that S is a pruned superthin κ -Kurepa subtree of <κ κ .Finally, define T to consist of all elements of S together with all elements t of <κ κ with the property that there exist s ∈ ∂S such that s ⊆ t , dom( s ) ∈ Lim and t ( α ) = 0 for all α ∈ dom( t ) \ dom( s ). Claim. T is a <κ -closed pruned κ -Kurepa subtree of <κ κ .Proof of the Claim. Note that for every t ∈ T \ S , there is a unique limit or-dinal α ≤ dom( t ) with t ↾ α ∈ ∂S . Since S is superthin, this shows that | ( T \ S ) ∩ α κ | < κ holds for all α < κ . Moreover, since S is a κ -Kurepa sub-tree of <κ κ , this directly implies that T is also a κ -Kurepa subtree of <κ κ . Next,note that for every t ∈ T \ S , we have t ( t ∪ {h dom( t ) , i} ∈ T . Since S ispruned, this shows that T is also pruned. Finally, assume that there is t ∈ ∂T withdom( t ) ∈ Lim. Then t / ∈ S and, by the definition of T , we have t ↾ α ∈ S for all α < dom( t ). But then t ∈ ∂S ⊆ T , a contradiction. (cid:3) By the above claim, Proposition 4.2 directly shows that [ T ] is a retract of κ κ . (cid:3) The properties of tree introduced in the next definition are studied in depth byBernhard K¨onig in [8].
Definition 4.4. (i) A tree of height λ is trivially coherent if it isomorphic toa subtree of <λ t with the property that t − { } is a finite set.(ii) A tree T is locally coherent if T <α is trivially coherent for every α < ht( T ). Proposition 4.5.
Let κ be a regular cardinal with λ ω < κ for all λ < κ . Thenevery locally coherent κ -Kurepa subtree of <κ κ is superthin.Proof. Let T be a locally coherent κ -Kurepa subtree of <κ κ and let α ∈ Lim ∩ κ .First, assume that cof( α ) = ω . Pick a cofinal sequence h α n | n < ω i in α .Given n < ω , the fact that T is a κ -Kurepa tree implies that λ n = | T ∩ α n κ | < κ .Moreover, since κ is regular, we know that λ = sup n<ω λ n < κ . But then theset [ T ∩ α κ ] has cardinality at most λ ω < κ and hence we can conclude that | ( T ∪ ∂T ) ∩ α κ | < κ .Next, assume that cof( α ) > ω . Fix a tree monomorphism π : T ∩ <α κ −→ <α π ( t ) − { } is finite for every t ∈ T ∩ <α κ . Given u ∈ [ T ∩ α κ ], we can now find minimal α u < α and N u < ω with the property that | π ( u ↾ ¯ α ) − { }| = N u holds for all α u ≤ ¯ α < α . In this situation, two elements u and u of [ T ∩ α κ ] are identical if and only if u ↾ α u = u ↾ α u holds. Inparticular, we also have | ( T ∪ ∂T ) ∩ α κ | ≤ | [ T ∩ α κ ] | < κ in this case. (cid:3) The following unpublished result of Donder shows that, in the constructible uni-verse, locally coherent Kurepa trees exist at all successor cardinals. This resultis proven by showing that the initial segments of the canonical Kurepa trees con-structed from the canonical morasses at successor cardinals in L (see [2, ChapterVIII, Section 2] and [13, Section 2]) satisfy the criterion for trivial coherency givenby [8, Lemma 2.17].
Theorem 4.6 (Donder) . Assume that
V = L holds. If κ is the successor of aninfinite cardinal, then there is a locally coherent κ -Kurepa tree.Proof of Theorem 1.7. Assume that V = L and let κ be an uncountable regularcardinal. If there is a cardinal µ with κ = µ + and cof( µ ) = ω , then Theorem1.1 directly implies that there is no κ -Kurepa subtree T of <κ κ with the propertythat [ T ] is a continuous image of κ κ . Moreover, if κ is inaccessible, then <κ κ -Kurepa subtree of <κ κ and the set [ <κ
2] = κ κ . Finally, if κ is the successor of a cardinal of uncountable cofinality, then λ ω < κ holds for all λ < κ and hence a combination of Lemma 4.3, Proposition 4.5 and Theorem 4.6shows that there is a κ -Kurepa subtree T of <κ κ with the property that [ T ] is aretract of κ κ . Since the GCH holds in L, the above observations provide the desiredcharacterization. (cid:3) ESCRIPTIVE PROPERTIES OF HIGHER KUREPA TREES 11
Our proofs of Theorem 1.9 and Theorem 1.10 will heavily rely on the followingproperties of inner models that were isolated by Hamkins in [4].
Definition 4.7.
Let µ be an infinite cardinal and let M be an inner model.(i) The pair ( M, V) has the µ -cover property if for every set x with x ⊆ M and | x | < µ , there is c ∈ M with c ⊆ x and | c | M < µ .(ii) The pair ( M, V) has the µ -approximation property if x ∈ M holds for everyset x with x ⊆ M and a ∩ x ∈ M whenever a ∈ M with | a | M < µ . Lemma 4.8.
Let M be an inner model such that the following statements hold forinfinite regular cardinals µ < κ with λ <µ < κ for all λ < κ : (i) The pair ( M, V) satisfies the µ -approximation property. (ii) κ is inaccessible in M . (iii) (2 κ ) M ≥ κ + .Then T = ( <κ M is a pruned superthin κ -Kurepa subtree of <κ κ .Proof. First, note that our second and third assumption directly imply that T isa pruned κ -Kurepa subtree of <κ κ . Fix α ∈ Lim ∩ κ . If cof( α ) < µ , then ourassumptions directly imply that | ∂T ∩ α κ | ≤ | [ T ∩ <α κ ] | < κ . In the other case, ifcof( α ) ≥ µ and x ∈ [ T ∩ <α κ ], then a ∩ x ∈ M holds for all a ∈ M with | a | M < µ and hence the µ -approximation property implies that x is an element of M . Thisargument shows that [ T ∩ <α κ ] ⊆ T holds for all α ∈ Lim ∩ κ with cof( α ) ≥ µ . Inparticular, this shows that ∂T ∩ α κ = ∅ holds for all such ordinals α . Since T is a κ -Kurepa subtree of <κ κ , these computations show that T is superthin. (cid:3) The proof of [10, Theorem 7.2] directly yields the following result needed for theproof of Theorem 1.9.
Theorem 4.9.
Let M be an inner model such that R * M and the pair ( M, V) has the ℵ -cover property. If κ is an uncountable regular cardinal with | (2 κ ) M | > κ and T = ( <κ M , then the set [ T ] is not a continuous image of κ κ .Proof of Theorem 1.9. Let κ be an inaccessible cardinal, let µ < κ be an uncount-able regular cardinal, let x be Add( ω, G be Col( µ, <κ )-generic over V[ x ]. Then all cofinalities less than or equal to µ and greater thanor equal to κ are preserved in V[ x, G ] and κ = ( µ + ) V[ x,G ] . Moreover, [4, Lemma13] shows that the pair (V , V[ x, G ]) has the ℵ -approximation and ℵ -cover prop-erty. Set T = ( <κ V . Since κ is inaccessible in V, (2 κ ) V = (2 κ ) V[ x,G ] and( λ ω ) V[ x,G ] = ( λ ω ) V < κ holds for all λ < κ , Lemma 4.8 shows that T is a prunedsuperthin κ -Kurepa subtree of <κ κ . In addition, since R V[ x,G ] * V, we can applyTheorem 4.9 to conclude that the set [ T ] is not a continuous image of κ κ in V[ x, G ].Finally, an application of Lemma 4.3 yields a κ -Kurepa subtree T of <κ κ in V[ x, G ]such that T ⊆ T and the set [ T ] is a retract of κ κ in V[ x, G ]. (cid:3) Proof of Theorem 1.10.
Let κ be a 2-Mahlo cardinal, let E denote the set of Mahlocardinals smaller than κ and let x be Add( ω, x ] andlet hh ~ P <α | α ≤ κ i , h ˙ P α | α < κ ii be a forcing iteration with Easton support suchthat the following statements hold whenever H is ~ P <α -generic over V[ x ] for some α < κ : • If α ∈ E , then ˙ P H = Col( α, α ) V[ x,H ] . • If α / ∈ E , then ˙ P H is the trivial partial order. Claim.
Forcing with ~ P <κ over V[ x ] preserves the inaccessibility of κ , the station-arity of E in κ and the regularity of all elements of E and of all regular cardinalsgreater than or equal to κ .Proof of the Claim. Let α ≤ κ be a Mahlo cardinal, let G be ~ P <κ -generic overV[ x ] and let H denote the filter on ~ P <α induced by G . Then [1, Proposition 7.13]shows that ~ P <α satisfies the α -chain condition in V[ x ] and [1, Proposition 7.12]implies that the induced tail forcing ˙ P H [ α,κ ) is <α -closed in V[ x, H ]. In particular, α is regular in V[ x, G ], stationary subset of α in V are stationary in V[ x, G ] and, if α < κ , then (2 <α ) V[ x,G ] = α . (cid:3) Let G be ~ P <κ -generic over V[ x ]. Since [1, Proposition 7.12] implies that ~ P <κ is σ -closed in V[ x ], we can apply [4, Lemma 13] to conclude that the pair (V , V[ x, G ])has the ℵ -approximation property. Set T = ( <κ V and work in V[ x, G ]. ThenLemma 4.8 implies that T is a superthin κ -Kurepa subtree of <κ κ . Let T be thetree constructed from T as in the proof of Lemma 4.3. Claim. If α ∈ E , then | T ∩ α κ | = α .Proof of the Claim. First, note that, if we repeat the construction from the proofof Lemma 4.3, then S = S . Fix α ∈ E . Then our forcing construction ensuresthat | T ∩ α κ | ≤ | (2 α ) V | = α holds. Next, notice that, if ¯ α ∈ Lim ∩ α , then ∂T ∩ ¯ α κ ⊆ ¯ α | ∂T ∩ ¯ α κ | ≤ <α = α . Finally, since the pair (V , V[ x, G ]) has the ℵ -approximationproperty and α is regular in V[ x, G ], we also know that ∂T ∩ α κ = ∅ . By thedefinition of T , these observations imply the statement of the claim. (cid:3) Since the proof of Lemma 4.3 shows that T is a κ -Kurepa subtree of <κ κ andthe set [ T ] is a retract of κ κ , the above claim shows that the tree possesses all ofthe desired properties. (cid:3) Isolated points
The following simple observation will be central for our investigation of retrac-tions of generalized Baire space onto the sets of cofinal branches of Kurepa trees.
Proposition 5.1.
Let κ be an uncountable regular cardinal and let r be a retractionfrom κ κ to a subset X of κ κ . If α < κ and A ∈ [ X ] <κ , then there is α < β < κ such that the following statements hold: (i) x ↾ β = y ↾ β for all x, y ∈ A with x = y . (ii) r [ N x ↾ β ] ⊆ N x ↾ β for all x, y ∈ A .Proof. We inductively construct a strictly increasing sequence h β n | n < ω i ofordinals in the interval ( α, κ ). Since | A | < κ , we can find α < β < κ with x ↾ β = y ↾ β for all x, y ∈ A with x = y . Next, assume that β n is already definedfor some n < ω . Given x ∈ A , we can then find β xn ∈ ( β n , κ ) with r [ N x ↾ β xn ] ⊆ N x ↾ β n .Set β n +1 = sup x ∈ A β xn < κ . Finally, define β = sup n<ω β n < κ . Given x ∈ A , thisconstruction ensures that r [ N x ↾ β n +1 ] ⊆ N x ↾ β n holds for all n < ω and this allowsus to conclude that r [ N x ↾ β ] ⊆ N x ↾ β holds. (cid:3) Using the above proposition, we now show that Kurepa trees that are retractscontain isolated cofinal branches.
ESCRIPTIVE PROPERTIES OF HIGHER KUREPA TREES 13
Proof of Theorem 1.11.
Let κ be an uncountable regular cardinal and let T be a κ -Kurepa subtree of <κ κ such that there is a continuous function r : κ κ −→ [ T ]with r ↾ [ T ] = id [ T ] and [ T ] does not have isolated points. Let µ ≤ κ be the leastcardinal with 2 µ ≥ κ .In the following, we inductively construct a strictly increasing continuous se-quence h α ( ξ ) < κ | ξ ≤ µ i of ordinals and a sequence h t s ∈ T ∩ α ( ξ ) κ ) | ξ ≤ µ, s ∈ ξ i such that the following statements hold for all ξ ≤ µ and s , s ∈ ξ ζ < ξ , then t s ↾ ζ = t s ↾ α ( ζ ).(ii) For all ζ ≤ ξ , we have t s ↾ α ( ζ ) = t s ↾ α ( ζ ) if and only if s ↾ ζ = s ↾ ζ .(iii) If ξ ∈ Lim, then t s = S ζ<ξ t s ↾ ζ .(iv) r [ N t s ] ⊆ [ T ] ∩ N t s .Set α (0) = 0 and t ∅ = ∅ . Next, assume that ξ < µ and the sequences h α ( ζ ) | ζ ≤ ξ i and h t s | ζ ≤ ξ, s ∈ ζ i are already constructed. Fix a sequence s in ξ
2. Sincewe have r [ N t s ] ⊆ [ T ] ∩ N t s = ∅ and [ T ] contains no isolated points, there are x s , x s ∈ N t s ∩ [ T ] with x s = x s . Set A ξ = { x si | s ∈ ξ , i < } . Then the minimal-ity of µ implies that | A ξ | < κ and hence we can apply Proposition 5.1 to α ( ξ ) and A ξ to find α ( ξ ) < α ( ξ + 1) < κ with the listed properties. Given s ∈ ξ i < t s ⌢ h i i = x si ↾ α ( ξ + 1). Finally, let ξ ≤ µ be a limit ordinal. Set α ( ξ ) = sup ζ<ξ α ( ζ ) and define t s = S ζ<ξ t s ↾ ζ for all s ∈ ξ
2. Given s ∈ ξ
2, we thenhave r [ N t s ] ⊆ \ ζ<ξ r [ N t s ↾ ζ ] ⊆ \ ζ<ξ ([ T ] ∩ N t s ↾ ζ ) = [ T ] ∩ N t s and, since ∅ 6 = r [ N t s ] ⊆ [ T ] ∩ N t s , this shows that t s ∈ T .Now, assume that κ is not inaccessible. Then µ < κ and the above constructionshows that κ ≤ µ = |{ t s | s ∈ µ }| ≤ | T ∩ α ( µ ) κ | , a contradiction.This shows that κ is inaccessible and κ = µ . Then the set C = { ξ < κ | α ( ξ ) = ξ is a cardinal } is closed and unbounded in κ . Given ξ ∈ C , the above construction ensures thatthe set T ∩ ξ κ has cardinality 2 ξ > ξ . In particular, the tree T is not stationaryslim. (cid:3) Our next goal is to show that every Kurepa tree can be thinned out to obtaina Kurepa tree without isolated branches. The proof of the next lemma is a directadaptation of a classical argument to higher cardinals.
Lemma 5.2.
Let κ be an uncountable regular cardinal and let T be a subtree of <κ κ with | [ T ] | > κ <κ . Then there is a subtree S of <κ κ such that S ⊆ T , | [ T ] | = | [ S ] | and [ S ] contains no isolated points.Proof. Set θ = κ <κ and let h C γ | γ < θ + i denote the unique sequence of closedsubsets of κ κ such that C = [ T ] and the following statements hold for all γ ≤ θ + :(i) If γ < θ + and I γ is the set of isolated points of C γ , then C γ +1 = C γ \ I γ .(ii) If γ is a limit ordinal, then C γ = T β<γ C β .Assume, towards a contradiction, that C γ = C θ + for every γ < θ + . Then I γ = ∅ for every γ < θ + . Given γ < θ + and x ∈ I γ , let α xγ denote the least α < κ with C γ ∩ N x ↾ α = { x } . Next, define K γ = { x ↾ α xγ ∈ <κ κ | x ∈ I γ } 6 = ∅ . Then we have K γ ∩ K ¯ γ = ∅ for all ¯ γ < γ < θ + = ( κ <κ ) + , a contradiction.The above computations show that there is a γ < θ + with C γ = C θ + and thisshows that the closed set C γ has no isolated points. Let S denote the canonicalsubtree of <κ κ with [ S ] = C γ . Then S ⊆ T and | [ S ] | = | [ T ] \ [ β<γ I β | = | [ T ] | , because for every β < θ + , the fact that I β consists of the isolated points of C β ,allows us to conclude that | I β | ≤ κ <κ < | [ T ] | . (cid:3) Using the above lemma, we are now able to prove the remaining results fromSection 1.3.
Proof of Theorem 1.12.
Let κ be an uncountable regular cardinal with κ <κ = κ and assume there exists a κ -Kurepa tree. If κ is inaccessible, then Theorem 1.4shows that there is a κ -Kurepa subtree T of <κ κ with the property that the set [ T ]is no a retract of κ κ . In the other case, if κ is not inaccessible, then our assumptionallows us to use Lemma 5.2 to find a κ -Kurepa subtree T of <κ κ with the propertythat the set [ T ] has no isolated points and Theorem 1.11 then shows that the set[ T ] is not a retract of κ κ . Finally, if there exists a stationary slim κ -Kurepa subtree T of <κ κ and S is the subtree of T produced by an application of Lemma 5.2, then S is also stationary slim, the set [ S ] has no isolated points and hence Theorem 1.11shows that the set [ S ] is not a retract of κ κ . (cid:3) Proof of Theorem 1.13.
Let κ be an uncountable regular cardinal, let S be a κ -Kurepa subtree of <κ κ and let c : κ κ −→ [ S ] be a continuous surjection. We let I denote the set of isolated points of [ S ] and we fix an injection ι : I −→ S with N ι ( z ) ∩ [ S ] = { z } for all z ∈ I . Define T to be the union of S and the set { t ∈ <κ κ | ∃ z ∈ I [ ι ( z ) ⊆ t ∧ |{ α ∈ dom( s ) \ dom( ι ( z )) | t ( α ) = z ( α ) }| < ω ] } . Then T is a subtree of <κ κ with | T ∩ α κ | ≤ | S ∩ α κ | + | α | + ω < κ for all α < κ .Moreover, it is easy to see that[ T ] = [ S ] ∪ { y ∈ κ κ | ∃ z ∈ I [ ι ( z ) ⊆ y ∧ |{ α < κ | y ( α ) = z ( α ) }| < ω ] } . This shows that T is a κ -Kurepa subtree of <κ κ with the property that the set[ T ] does not contain isolated points and, if S is stationary slim, then T is alsostationary slim.For each z ∈ I , the set N ι ( z ) ∩ [ T ] has cardinality κ and we can fix an enumeration h y z ( α ) | α < κ i of this set. Moreover, we pick an injection ρ : I −→ <κ κ with theproperty that c [ N ρ ( z ) ] ⊆ N ι ( z ) holds for all z ∈ I . Then c [ N ρ ( z ) ] = { z } holds for all z ∈ I . Set O = S { N ρ ( z ) | z ∈ I } and let d : κ κ −→ [ T ] denote the unique functionwith d ↾ ( κ κ \ O ) = c ↾ ( κ κ \ O ) and d ( x ) = y c ( x ) ( x (dom( ρ ( c ( x ))))) for all x ∈ O .Then d is a surjection. Claim.
The map d is continuous.Proof of the Claim. Fix x ∈ κ κ and β < κ .First, assume that x ∈ O . Then d [ N x ↾ (dom( ρ ( c ( x )))+1) ] = { d ( x ) } ⊆ N d ( x ) ↾ β .Next, assume that x / ∈ O and c ( x ) ∈ I . Since c ( x ) is isolated in T and x / ∈ O ,there is α < κ with N x ↾ α ∩ N ρ ( c ( x )) = ∅ and c [ N x ↾ α ] = { c ( x ) } . Then N x ↾ α ∩ O = ∅ and we can conclude that d [ N x ↾ α ] = c [ N x ↾ α ] = { c ( x ) } ⊆ N d ( x ) ↾ β . ESCRIPTIVE PROPERTIES OF HIGHER KUREPA TREES 15
Finally, assume that x / ∈ O and c ( x ) / ∈ I . Then there is α < κ with c [ N x ↾ α ] ⊆ N c ( x ) ↾ β . Fix u ∈ N x ↾ α ∩ O . Then c ( u ) ∈ N ι ( c ( u )) ∩ N c ( x ) ↾ β = ∅ and this implies thatthe sequences ι ( c ( u )) and c ( x ) ↾ β are comparable. But then c ( x ) ↾ β ( ι ( c ( u )),because ι ( c ( u )) ⊆ c ( x ) ↾ β would imply that c ( x ) ∈ N c ( x ) ↾ β ∩ [ T ] ⊆ N ι ( ρ ( c ( u ))) ∩ [ T ] = { c ( u ) } ⊆ I . This shows that d ( u ) = y c ( u ) ( u (dom( ρ ( c ( u ))))) ∈ N ι ( c ( u )) ⊆ N c ( x ) ↾ β .Since c ( x ) = d ( x ), these computations show that d [ N x ↾ α ] ⊆ N d ( x ) ↾ β holds. (cid:3) Finally, assume that the set [ T ] is a retract of κ κ . Then Theorem 1.11 showsthat κ is inaccessible and S is not stationary slim. Define U = { u ∈ <κ | ( ∃ α ∈ dom( u ) u ( α ) = 1) −→ u (0) = 1 } . Then U is a κ -Kurepa subtree of <κ κ , the set [ U ] is a retract of κ κ and the uniqueelement x of κ κ with x (0) = 0 is an isolated point of [ U ]. Let W be the κ -Kurepasubtree of <κ κ obtained from U through the above construction. Then there is0 < β < κ such that the set [W] is equal to { x ∈ κ | x (0) = 1 } ∪ { x ∈ κ κ | ∀ α < β x ( α ) = 0 ∧ |{ α < κ | x ( α ) = 0 }| < ω } . Assume, towards a contradiction, that r : κ κ −→ [ W ] is a map witnessing that[ W ] is a retract of κ κ . We now inductively construct a sequence h x n | n < ω i ofelements of [ W ] and a strictly increasing sequence h β n | n < ω i of ordinals below κ such that x n ( β n ) = 0 and x n ↾ β n +1 = x n +1 ↾ β n +1 for all n < ω . Let x be an arbitrary element of [ W ] with x (0) = 0 and x ( β ) = 0. If x n and β n are already constructed, then we can apply Proposition 5.1 to find β n +1 ∈ ( β n , κ )with r [ N x n ↾ β n +1 ] ⊆ N x n ↾ β n +1 and we pick x n +1 ∈ [ W ] with x n ↾ β n +1 ⊆ x n +1 and x n +1 ( β n +1 ) = 0. Pick x ∈ κ κ with x n ↾ β n +1 ⊆ x for all n < ω . Then x n ↾ β n +1 ⊆ r ( x ) for all n < ω and hence r ( x )( β n ) = x n ( β n ) = 0 holds for all n < ω . Since r ( x )( α ) = 0 holds for all α < β , this shows that r ( x ) / ∈ [ W ], acontradiction. (cid:3) Open questions
We end this paper with a compilation of questions left open by the above results.First, note that Theorem 1.1 shows that by forcing with Add( ω, ω ) over L, weproduce a model of set theory in which there exist ℵ -Kurepa trees and for everysuch tree T , the set [ T ] is not a continuous image of ω ω . In addition, Theorems1.6 and 1.7 show that, in L, there exist ℵ -Kurepa trees T and T such that theset [ T ] is a continuous image of ω ω and the set [ T ] is not a continuous image of ω ω . Therefore, there is only one constellation whose consistency is not settled byour results: Question 6.1.
Is it consistent with the axioms of
ZFC that there are ℵ -Kurepatrees and for every such tree T , the set [ T ] is a continuous image of ω ω ? Next, note that the only way to apply the above results to obtain models ofZFC that contain ℵ -Kurepa trees and have the property that no such tree is acontinuous image of ω ω is to consider models in which CH fails. Thus, it isnatural to ask the following question: Question 6.2.
Does CH together with the existence of an ℵ -Kurepa tree implythat there exists an ℵ -Kurepa subtree T of <ω ω with the property that the set [ T ] is: (i) a continuous image of ω ω ? (ii) a retract of ω ω ? Finally, the Kurepa trees constructed in the proofs of the results presented inSection 1.2 all arise from modifications of canonical Kurepa trees whose existenceis ensured by the assumptions of these results. Therefore, it is also interesting tostudy the descriptive properties of these canonical Kurepa trees themselves.
Question 6.3.
Let κ denote an uncountable regular cardinal with the property that µ ω < κ holds for all µ < κ . (i) If T is the canonical κ -Kurepa subtree of <κ κ constructed from a ♦ + κ -sequence (see [9, Chapter II, Section 7] ), is the set [ T ] a continuous imageof κ κ ? (ii) If T is the canonical κ -Kurepa subtree of <κ κ constructed from a ( κ, -morass (see [13, Section 1.2] ) or a simplified morass (see [14] ), is the set [ T ] a continuous image of κ κ ? References
1. James Cummings,
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