@ Appl. Gen. Topol. 15, no. 2(2014), 137-146doi:10.4995/agt.2014.3109 c© AGT, UPV, 2014 Asymptotic structures of cardinals Oleksandr Petrenko a, Igor Protasov a and Sergii Slobodianiuk b a Department of Cybernetics, Kyiv National University, Ukraine (opetrenko72@gmail.com, i.v.protasov@gmail.com) b Department of Mechanics and Mathematics, Kyiv National University, Ukraine (slobodianiuk@yandex.ru) Abstract A ballean is a set X endowed with some family F of its subsets, called the balls, in such a way that (X, F) can be considered as an asymp- totic counterpart of a uniform topological space. Given a cardinal κ, we define F using a natural order structure on κ. We characterize balleans up to coarse equivalence, give the criterions of metrizability and cellularity, calculate the basic cardinal invariant of these balleans. We conclude the paper with discussion of some special ultrafilters on cardinal balleans. 2010 MSC: 54A25; 05A18. Keywords: cardinal balleans; coarse equivalence; metrizability; cellularity; cardinal invariants; ultrafilter. 1. Introduction Following [15] we say that a ball structure is a triple B = (X, P, B), where X, P are non-empty sets and, for every x ∈ X and α ∈ P , B(x, α) is a subset of X which is called a ball of radius α around x. It is supposed that x ∈ B(x, α) for all x ∈ X and α ∈ P . The set X is called the support of B, P is called the set of radii. Given any x ∈ X, A ⊆ X, α ∈ P , we set B∗(x, α) = {y ∈ X : x ∈ B(y, α)}, B(A, α) = ⋃ a∈A B(a, α). Received 15 May 2013 – Accepted 15 February 2014 http://dx.doi.org/10.4995/agt.2014.3109 I. V. Protasov, O. Petrenko and S. Slobodianiuk A ball structure B = (X, P, B) is called a ballean if (1) for any α, β ∈ P , there exist α′, β′ such that, for every x ∈ X, B(x, α) ⊆ B∗(x, α′), B∗(x, β) ⊆ B(x, β′); (2) for any α, β ∈ P , there exists γ ∈ P such that, for every x ∈ X, B(B(x, α), β) ⊆ B(x, γ); (3) for any x, y ∈ X, there exists α ∈ P such that y ∈ B(x, α). A ballean on X can also be determined in terms of entourages of the diagonal ∆X of X × X, in this case it is called a coarse structure [16]. For balleans as counterparts of uniform topological spaces see [15, Chapter 1]. Let B = (X, P, B), B′ = (X′, P ′, B′) be balleans. A mapping f : X → X′ is called a ≺-mapping if, for every α ∈ P , there exists α′ ∈ P ′ such that, for every x ∈ X, f(B(x, α)) ⊆ B′(f(x), α′). If there exists a bijection f : X → X′ such that f and f−1 are ≺-mappings, B and B′ are called asymorphic and f is called an asymorphism. For a ballean B = (X, P, B), a subset Y ⊆ X is called large if there is α ∈ P such that X = B(Y, α). A subset V of X is called bounded if V ⊆ B(x, α) for some x ∈ X and α ∈ P . Each non-empty subset Y ⊆ X determines a subballean BY = (Y, P, BY ) where BY (y, α) = Y ∩B(y, α). We say that B and B′ are coarsely equivalent if there exist large subset Y ⊆ X and Y ′ ⊆ X′ such that the subballeans BY and B ′ Y ′ are asymorphic. Given a cardinal κ > 0 and ordinals x, α ∈ κ, we put −→ B(x, α) = [x, x+α] = {y ∈ κ : x 6 y 6 x+α}, ←− B(x, α) = {y ∈ κ : x ∈ [y, y+α]}, ←→ B (x, α) = −→ B(x, α) ∪ ←− B(x, α), so we have got three ball structures −→ κ = (κ, κ, −→ B),←−κ = (κ, κ, ←− B),←→κ = (κ, κ, ←→ B ). It is easy to see that, for κ > 1, the ball structures −→κ and ←−κ do not satisfy (1), so −→κ and ←−κ are not balleans, but ←→κ is a ballean for each κ > 0. This ballean ←→κ is called a cardinal ballean on κ. If κ is finite then the ballean ←→κ is bounded and any two bounded balleans are coarsely equivalent, so in what follows all cardinal balleans are supposed to be infinite. In section 2, we characterize cardinal balleans up to coarse equivalence. The criterions of metrizability and cellularity are given in section 3. In section 4 we study the basic cardinal invariants of cardinal ballean. In section 5 we consider the action of the combinatorial derivation on cardinals. We conclude the paper with discussion in section 6 of some special ultrafilters on cardinal balleans. c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 2 138 Asymptotic structures of cardinals 2. Coarse equivalence For a ballean B = (X, P, B), we use a preordering < on P defined by the rule: α < β if and only if B(x, α) ⊆ B(x, β) for each x ∈ X. A cofinality cfB is the minimal cardinality of cofinal subsets of P . A ballean B is called ordinal if there exists a well-ordered by < cofinal subset of P . If B is ordinal then B is asymorphic to B′ = (X, κ, B′) for some cardinal κ. Each ballean coarsely equivalent to some ordinal ballean is ordinal. Clearly, ←→ κ is ordinal for each κ. We say that a family F of subsets of X is uniformly bounded in B if there exists α ∈ P such that, for every F ∈F, there is x ∈ X such that F ⊆ B(x, α). Theorem 2.1. A ballean B(X, κ, B) is coarsely equivalent to ←→κ if there exists x0 ∈ X and γ ∈ κ such that (i) B(x0, α + γ) \B(x0, α) 6= ∅ for each α ∈ κ; (ii) for every β ∈ κ, the family {B(x0, α+β)\B(x0, α) : α ∈ κ} is uniformly bounded in B. Proof. We choose a subset A ⊆ κ such that [α, α + γ) ∩ [α′, α′ + γ) = ∅ for all distinct α, α′ ∈ A and κ = ∪α∈A[α, α + γ). For each α ∈ A, we use (i) to pick some element x(α) ∈ B(x0, α + γ) \ B(x0, α) and note that the set L = {x(α) : α ∈ A} is large in B. Then we denote by f : A → κ the natural well-ordering of A as a subset of κ and, for each α ∈ A, put h(x(α)) = f(α). Clearly, h is a bijection from L to κ. By (ii), h is an asymorphism between L and κ. Since L is large in B, we conclude that B and ←→κ are coarsely equivalent. � For distinct cardinals κ and κ′, the balleans ←→κ and ←→ κ′ are not coarsely equivalent because (see Theorem 4.1 (i)) each large subset of ←→κ (resp. ←→ κ′ has cardinality κ (resp. κ′), so there are no bijections (in particular, asymorphisms) between large subsets of ←→κ and ←→ κ′ . 3. Metrizability and cellularity Each metric space (X, d) defines a metric ballean (X, R+, Bd), where Bd(x, r) = {y ∈ X : d(x, y) 6 r}. By [15, Theorem 2.1.1], for a ballean B, the following conditions are equivalent: cfB 6 ℵ0, B is asymorphic to some metric ballean, B is coarsely equivalent to some metric ballean. In this case B is called metrizable. Applying this criterion, we get Theorem 3.1. For a cardinal κ, the following conditions are equivalent: (i) cfκ = ℵ0; (ii) ←→κ is asymorphic to some metric ballean; (iii) ←→κ is coarsely equivalent to some metric ballean. Given an arbitrary ballean B = (X, P, B), x, y ∈ X and α ∈ P , we say that x and y are α-path connected if there exists a finite sequence x0, . . . , xn, c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 2 139 I. V. Protasov, O. Petrenko and S. Slobodianiuk x0 = x, xn = y such that xi+1 ∈ B(xi, α) for each i ∈ {0, . . . , n − 1}. For any x ∈ X and α ∈ P , we set B�(x, α) = {y ∈ X : x, y are α-path connected}. The ballean B� = (X, P, B�) is called a cellularization of B. A ballean B is called cellular if the identity mapping id : X → X is an asymorphism between B and B�. By [15, Theorem 3.1.3], B is cellular if and only if asdimB = 0, where asdimB is the asymptotic dimension of B. By [15, Theorem 3.1.1], a metric ballean is cellular if and only if B is asymorphic to a ballean of some ultrametric space. Theorem 3.2. For a cardinal κ, the ballean ←→κ is cellular if and only if κ > ℵ0. Proof. The ballean ←→ ℵ0 is not cellular because B �(0, 1) = ℵ0 so ←→ ℵ0 � is bounded but ←→ ℵ0 is unbounded. Assume that κ > ℵ0, fix an arbitrary α ∈ κ and note that B�(x, α) ⊆ ⋃ n∈ℵ0 ←→ B (x, nα) ⊆ ←→ B (x, γ), where γ = supn∈ℵ0 nα. Since κ > ℵ0, we have γ ∈ κ. Hence, id : κ → κ is an asymorphism between ←→κ and ←→κ �. � 4. Cardinal invariants Given a ballean B = (X, P, B), a subset A of X is called • large if X = B(A, α) for some α ∈ P ; • small if X \B(A, α) is large for every α ∈ P ; • thick if, for every α ∈ P , there exists a ∈ A such that B(a, α) ⊆ A; • thin if, for every α ∈ P , there exists a bounded subset V of X such that B(a, α) ∩B(a′, α) = ∅ for all distinct a, a′ ∈ A\V . We note that large, small, thick and thin subsets can be considered as as- ymptotic counterparts of dense, nowhere dense, open and uniformly discrete subsets subsets of a uniform topological space. The following cardinal characteristics of a ballean B were introduced and studied in [1], [10], [11], see also [15, Chapter 9]. All these characteristics are invariant under asymorphisms. denB = min{|L|: L is a large subset of X}, resB = sup{|F| : F is a family of pairwise disjoint large subsets of X}, coresB = min{|F| : F is a covering of X by small subsets}, thickB = sup{|F| : F is a family of pairwise disjoint thick subsets of X}, thinB = min{|F| : F is a covering of X by thin subsets}, spreadB = sup{|Y |B: Y is a thin subset of X}, where |Y |B = min{|Y \ V | : V is a bounded subset of X}. To prove Theorem 4.1, we need some ordinal arithmetics (see [2]). An ordinal γ, γ 6= 0 is called additively indecomposable if one of the following equivalent statements holds c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 2 140 Asymptotic structures of cardinals • α + β < γ for any ordinal α, β < γ; • α + γ = γ for every ordinal α < γ; Every ordinal α > 0 can be written uniquiely in the Cantor normal form α = n1 · γ1 + . . . + nk ·γk, where γ1 > . . . > γk are additively indecomposable, n1, . . . , nk are natural numbers. We put |α| = |{β : β < α}|, ||α|| = n1 + . . . + nk, r(α) = γk. Theorem 4.1. For every cardinal κ, the following statements hold (i) den←→κ = spread←→κ = κ; (ii) res←→κ = κ and ←→κ can be partitioned in κ large subsets; (iii) cores←→κ = ℵ0; (iv) thick←→κ = κ and ←→κ can be partitioned in κ thick subsets; (v) thin←→κ = cfκ if κ is a limit cardinal and thin κ+ = κ. Proof. (i) For a large subset L of X, we pick α ∈ P such that X = B(L, α), observe that |B(x, α)| 6 2|α| so |X| 6 |L|(2|α|) and |X| = |L|. Since the ballean ←→κ is ordinal, by [1, Theorem 2.3], spreadB = denB and there is a thin subset Y of X such that |Y |B = |Y |. (ii) For κ = ℵ0, we identify κ \{0} with the set N of natural numbers and partition N = ∪n∈ℵ02 nN, where N is the set of all odd numbers. Since each subset 2nN is large in ℵ0, we get a desired statement. Assume that κ > ℵ0, we have |L| = κ, so it suffices to show that each subset L(γ) is large in κ. Then ←→ B (α, γ)∩ L(γ) 6= ∅ for each α ∈ κ. (iii) The statement is trivial for κ = ℵ0 because each singleton in κ is small. Assume that κ > ℵ0 and, for each n ∈ N, put Sn = {α ∈ κ : ||α|| = n}. We show that, for all n ∈ N and γ ∈ κ, X \ ←→ B (Sn, γ) is large so Sn is small. Since κ > ℵ0, we have |Γ| = κ| and Γ is cofinal in κ. Thus, we may suppose that γ ∈ Γ. We take any x ∈ Sn, y ∈ ←→ B (x, γ) and note that if y has a member m ·γ in its Cantor normal form then m 6 n + 1. On the other hand, if k ·γ is the last member of x + (n + 2) · γ in its Cantor normal form then k > n + 2. Therefore ←→ B (x, (n + 2) ·γ)∩ (κ\B(Sn, γ)) 6= ∅. It follows that κ\ ←→ B (Sn, γ) is large. (iv) Since the ballean ←→κ is ordinal, by [11, Theorem 3.1], thick←→κ = den←→κ and there is a disjoint family of cardinality den←→κ consisting of thick subsets, so we can apply (ii). (v) Assume that γ+ < κ but κ can be partitioned into γ thin subsets κ = ∪β<γTβ. By the definition of thin subsets, for each β < γ, there is x(β) ∈ κ such that |[α, α + γ+] ∩ Tβ| 6 1 for each α > x(β). We choose x ∈ κ such that x > x(β) for each β < γ. Then |[x, x + γ+] ∩ ∪β<γTβ| 6 γ. Since |[x, x+γ+]| = γ+, we have |[x, x+γ+]\∪β<γTβ 6= ∅, contradicting κ = ∪β<γTβ. c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 2 141 I. V. Protasov, O. Petrenko and S. Slobodianiuk If κ is a limit cardinal, by above paragraph, κ can not be partitioned into < cfκ thin subsets. Since each bounded subset is thin, we have thin κ = cfκ. To conclude the proof, we partition κ+ into κ thin subsets. Let {γα : α < κ+} be the numeration of all additively indecomposable ordinals from [κ, κ+). For α ∈ κ+, we fix some bijection fα : γ → [γα, γα+1) and, for each λ ∈ κ, put Tλ = {fα(λ) : α ∈ κ +}. Then [κ, κ+) = ∪λ<κTλ. Clearly [0, κ) is bounded (and so thin) and if x ∈ Tλ and x > γα then ←→ B (x, γα)∩ Tλ = {x}, so each subset Tλ is thin. � It is worth to be marked that a set Γ of all additively indecomposable ordinals is not only small (Γ = S1) but also thin in ←→ κ , and so ”very small” in asymptotic sense. On the other hand, for κ > ℵ0, Γ is unbounded and closed in the order topology on κ, and so ”very large” in topological sense. Let B = (X, P, B) be a ballean and α ∈ P . Following [14], we say that a subset Y ⊆ X has asymptotically isolated balls if, for every β > α, there is y ∈ Y such that B(y, β) \ B(y, α) = ∅. If Y has asymptotically isolated balls for some α ∈ P , we say that Y has asymptotically isolated balls. A ballean B is called asymptotically scattered if each unbounded subset of X has asymptotically isolated balls. A subset Y ⊆ X is called asymptotically scattered if the subballean BY is asymptotically scattered. The scattered number of B is the cardinal scatB = {min |F| : F is a covering of X by asymptotically scattered subsets}. Question 4.2. For each κ, determine or evaluate scat←→κ . We say that a ballean B is weakly asymptotically scattered if, for every un- bounded subset Y of X, there is α ∈ P such that, for every β ∈ P , there exists y ∈ Y such that (B(y, β)\B(y, α))∩X = ∅. A subset Y of X is called weakly asymptotically scattered if the subballean BY is weakly asymptotically scat- tered. For κ > ℵ0, each small subset Sn from the proof of Theorem 4.1 (iii) is weakly asymptotically scattered, so ←→κ can be partitioned into ℵ0 weakly asymptotically scattered subsets. We note that for n > 2, Sn is not asymptot- ically scattered. Let γ be a cardinal. Following [13], we say that a ballean B = (X, P, B) is γ-extraresolvable if there exists a family F of large subsets of X such that |F| = γ and F ∩ F ′ is small whenever F, F ′ are distinct members of F. The extraresolvability of B is the cardinal exresB = {sup γ : B is γ-extraesolvable}. By [13, Theorem 4], exres ←→ ℵ0 = ℵ0. Question 4.3. For each κ, determine or evaluate exres κ. Under some additional to ZFC assumptions, an existence of ℵ1-Kurepa tree (see [6, p. 74]), we prove that exres ←→ ℵ1 > ℵ1. Recall that a poset (T, <) is c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 2 142 Asymptotic structures of cardinals a tree if for any t ∈ T the set sub(t) = {x ∈ T : x < t} is well-ordered. For any t ∈ T , L(t) denotes an ordinal type of sub(t) and L(T ) = sup{L(t) + 1 : t ∈ T}. A tree T is a κ-tree if L(T ) = κ and |Levα(T )| < κ, α < L(T ) where Levα(T ) = {t ∈ T : L(t) = α}. A κ-tree T is called κ-Kurepa if there exists κ+ maximal chains of cardinality κ in T . Now let T be an ℵ1-Kurepa tree. Fix any bijection fα : Levα(T ) → [αω; (α + 1)ω), α < ℵ1, and define f = ⋃ fα, Bi = f −1(Ci), i < ℵ2, where Ci is a maximal chain in T . Then every Bi is large and, for any i 6= j, |Bi ∩ Bj| < ℵ1. Question 4.4. In ZFC, does there exist a cardinal κ such that exres←→κ > κ? 5. The combinatorial derivation Given a subset A of κ, we denote ∆(A) = {α ∈ κ : A + α ∩ A is unbounded in κ}, A + α = {β + α : β ∈ A}, and say that ∆ : Pκ → Pκ is the combinatorial derivation. For a group version of ∆ and motivation of this definition see [12]. Theorem 5.1. If a subset A of κ is not small and κ is regular then ∆(A) is large. Proof. Since A is not small, there exists λ ∈ κ such that ←→ B (A, λ) is thick. We show that κ = ←→ B (∆(A), λ + λ). We take an arbitrary δ ∈ κ. Since ←→ B (A, λ) is thick, for each α ∈ κ, we can choose xα ∈ A, xα > α such that [xα, xα + δ + λ + λ] ⊂ ←→ B (A, λ). Since xα+δ+λ ∈ ←→ B (A, λ), there exists yα ∈ [δ, δ+λ+λ] such that xα+yα ∈ A. The set X = {xα : α ∈ κ} is cofinal in κ and |[δ, δ + λ + λ]| < κ. Since κ is regular, we can choose a cofinal subset X′ ⊆ X and y ∈ [δ, δ + λ + λ] such that yα = y for each xα ∈ Y . Hence, y ∈ ∆(A) and δ ∈ ←→ B (y, λ + λ) so κ = ←→ B (∆(A), λ + λ). � Corollary 5.2. For every finite partition of a regular cardinal κ = A1∪. . .∪An, there exists i ∈{1, . . . , n} such that ∆(Ai) is large. Proof. It suffices to note that at least one cell of the partition is not small. � By Theorem 4.1 (iii), each subset Sn = {α ∈ κ : ||α|| = n} of κ is small. It is clear that ∆(Sn) ⊆{0}∪S1 ∪ . . .∪Sn. Hence, ∆(Sn) is small and Corollary does not hold for countable partition even if κ is arbitrary large. Question 5.3. Is Theorem 5.1 true for all singular cardinals? By the definition of ∆, ∆(A) = 0 for each thin subset A of κ. Theorem 5.4. Let κ be a regular cardinal, A ⊆ κ and 0 ∈ A. Then there exist two thin subsets X, Y of κ such that ∆(X ∪ Y ) = A. c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 2 143 I. V. Protasov, O. Petrenko and S. Slobodianiuk Proof. We enumerate A = {aα : α < |A|}, put aα = 0 for each α ∈ κ, α > |A|, and bα = max{α, aα} for every α ∈ κ. We denote x00 = 0. Since κ is regular, for each α ∈ κ, we can choose inductively an increasing sequence (xαβ)β6α in κ such that (1) xα0 > supβ<α xββ + bα + bα; (2) xαβ+1 > xαβ + bα + bα, β < α. After κ steps, we denote X = {xαβ : β 6 α < κ}, Y = {xαβ + aβ : β 6 α < κ}. By (1) and (2), X and Y are thin and ∆(X ∪ Y ) = A. � Question 5.5. Is Theorem 5.4 true for all singular cardinals? For a cardinal κ, we denote by κ# the family of all ultrafilters on κ whose members are unbounded in κ. We say that a subset A of κ is sparse if, for every U ∈ κ, the set {α ∈ κ : A ∈ U + α} is bounded in κ, where U + α is an ultrafilter with the base {U + α : U ∈ U}. Clearly, the family of all sparse subsets of κ is an ideal in the Boolean algebra Pκ. If A is thin (in particular, A is bounded) then |{α ∈ κ : A ∈U + α}| 6 1 so A is sparse. We use sparse subsets to characterize strongly prime ultrafilters in κ#. We endow κ with the discrete topology, denote by βκ the Stone-Čech compactifi- cation of κ and use the universal property of βκ to extend the addition + from κ to βκ in such a way that, for each α ∈ κ, the mapping x 7→ x + α : βκ → βκ is continuous, and, for each U ∈ βκ, the mapping x 7→ U + x : βκ → βκ is continuous (see [4, Chapter 4]. To describe a base for the ultrafilter U +V, we take any element V ∈ V and, for every x ∈ V , choose some element Ux ∈ U. Then ∪x∈V (Ux + x) ∈ U + V, and the family of subsets of this form is a base for U +V. We note that κ# is a subsemigroup of βκ and say that an ultrafilter U ∈ κ# is strongly prime if U is not in the closure of κ# + κ#. With above definitions, we get the following characterization. Theorem 5.6. An ultrafilter U ∈ κ# is strongly prime if and only if some member U ∈U is sparse in κ. Question 5.7. What can be said about inclusions between families of sparse and asymptotically scattered subsets of κ? Question 5.8. Is a subset A of κ sparse provided that ∆(A) is sparse? 6. Around T -points A free ultrafilter U on an infinite cardinal κ is called uniform if |U| = κ for each U ∈ U. We say that a uniform ultrafilter U on κ is a Tκ-point if, for each minimal well-ordering < of κ, some member of U is thin in the ballean ((κ, <), (κ, <), ←→ B ). We begin with discussion of Tℵ0-points. Let G be a transitive group of permutations of κ. We consider a ballean B(G, κ) = (κ, [G]<ω, B), where [G]<ω is the family of all finite subsets of G and B(x, F) = {x}∪ F(x) for each x ∈ X and F ∈ [G]<ω, F(x) = {f(x), f ∈ F}. c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 2 144 Asymptotic structures of cardinals A free ultrafilter U on ℵ0 is said to be a T -point if, for every countable transitive group G of permutations of ℵ0, some member of U is thin in B(G,ℵ0). By [8, Theorem 1], every P-point and every Q-point (in βω) are T -points. Under CH, there exist P- and Q-points but it is unknown [3, Question 25] whether in each model of ZFC there exists either P-point or Q-point. By [5] and [7], this is so if c 6 ℵ2. By [8, Proposition 4], under CH, there exist a T -point which is neither Q-point nor weak P-point. It is an open question [9, p. 348] whether T -points exist in ZFC without additional assumptions. Theorem 6.1. Every T -point is a Tℵ0-point. Proof. A ballean B = (X, P, B) is called uniformly locally finite if, for each α ∈ P , there exists a natural number n such that |B(x, α)| 6 n for every x ∈ X. By [9, Theorem 9], for X = ℵ0 and a free ultrafilter U on X, the following statements are equivalent (i) U is a T -point; (ii) for every metrizable uniformity locally finite ballean B on X, some member of U is thin in B; (iii) for every sequence (Fn)n∈ω of uniformly bounded coverings of X, there exists U ∈U such that, for each n ∈ ω, |F ∩ U| 6 1 for all but finitely many F ∈ Fn. A covering F of ω is uniformly bounded if there is m ∈ ℵ0 such that, for each x ∈ X, |st(x,F)| 6 m, st(x,F) = ⋃ {F ∈ F : x ∈ F}. The equivalence (i) ⇔ (ii) implies that every T -point is a Tℵ0-point because each ballean ((ℵ0, <), (ℵ0, <), ←→ B ) is metrizable and uniformly locally finite. � Question 6.2. Does there exist a Tℵ0-point but not a T -point? Question 6.3. Does there exist a Tℵ0-point in ZFC? Let us call a free ultrafilter U on ℵ0 to be S-point if, for every minimal well ordering < of ℵ0, some member of U is small in the ballean ((ℵ0, <), (ℵ0, < ), ←→ B ). Question 6.4. Does there exist an S-point in ZFC? Theorem 6.5. In ZFC, for every uncountable regular cardinal κ, there exists a Tκ-point. Proof. We say that a uniform ultrafilter U on κ is a club-ultrafilter if U contains a family of all closed unbounded subsets of κ. By [17], U is a Q-point, i.e. U is selective with respect to any partition of κ into the cells of cardinality < κ. Now we fix some club-ultrafilter U on κ and prove that U is a Tκ-point. Let < be a minimal well-ordering of κ defined by some bijection h : κ → (κ, <). We partition inductively (κ, <) = ∪α∈(κ,<)[xα, xα + γα) into consecutive intervals, where Γ = {γα : α ∈ (κ, <)} is the set of all additively indecomposable ordi- nals of (κ, <). By above paragraph, some member U ∈ U meets each subset h−1([xα, xα+γα)) in at most one point. By the choice of intervals [xα, xα+γα), U is thin in the ballean ((κ, <), (κ, <), ←→ B ). � c© AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 2 145 I. V. Protasov, O. Petrenko and S. Slobodianiuk Question 6.6. 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