THE RUDIN–KEISLER ORDERING OF P-POINTS UNDER b = c

. M.E.Rudin(1971)proved,underCH,thatforeachP-point p thereexistsaP-point q strictly RK-greater than p . This result was proved under p = c by A. Blass (1973), who also showed that each RK-increasing (cid:2) -sequenceofP-pointsisupperboundedbyaP-point,andthatthereisanorderembedding of the real line into the class of P-points with respect to the RK-ordering. In this paper, the results cited above are proved under the (weaker) assumption that b = c . A. Blass asked in 1973 which ordinals can be embedded in the set of P-points, and pointed out that such an ordinal cannot be greater than c + . In this paper it is proved, under b = c , that for each ordinal α < c + , there is an order embedding of α into P-points. It is also proved, under b = c , that there is an embedding of the long line into P-points.

After a scrutiny of mechanisms underlying our proofs, we introduce an apparently new cardinal invariant q, the use of which enables us to weaken the set-theoretic assumptions of most of our results. Finally, we show that q is an instance of a general method of constructing useful variants of cardinal invariants.
In a recent paper, D. Raghavan and S. Shelah [8] proved (under p = c) that there is an order-embedding of P( )/fin into the set of P-points ordered by ≥ RK , and gave a short review of earlier results concerning embeddings of different orders into the class of P-points.
A. Blass also asked [1,Question 4] which ordinals can be embedded in the set of P-points, and pointed out that such an ordinal cannot be greater than c + . We show that under b = c each ordinal less than c + is order-embeddable into P-points. A recent paper by B. Kuzeljević and D. Raghavan [6] answers the question of the embedding of ordinals into P-points under MA. §2. Tools. A free ultrafilter u is a P-point if and only if, for each partition (V n ) n< of , there exists a set U ∈ u such that either U ⊂ V n for some n < or else U ∩ V n is finite for all n < . A filter F is said to be Rudin-Keisler greater (RK-greater) than a filter G (written as F ≥ RK G) if there exists a map h such that G ∈ G if and only if h -1 (G) ∈ F. Let be a partition of a subset of into infinite sets. A filter K is called a contour if there exists a partition W such that K ∈ K if and only if there is a cofinite set I ⊂ such that K ∩ W n is cofinite on W n for each n ∈ I . We call K a contour of W, and denote K = W. 3 A fundamental property used in the present paper is the following reformulation of [11, Proposition 2.1].

Proposition 2.1. A free ultrafilter is a P-point if and only if it does not include a contour.
As usual, c denotes the cardinality of the continuum. If f, g ∈ , then we say that f dominates g (and write f ≥ * g) if f(n) ≤ g(n) for almost all n < . We say that a family F of functions is unbounded if there is no g ∈ that dominates all functions f ∈ F. The minimal cardinality of an unbounded family is the bounding number b. We also say that a family F ⊂ is dominating if, for each g ∈ , there is some f ∈ F that dominates g. The dominating number d is the minimal cardinality of a dominating family. The pseudointersection number p is the minimal cardinality of a free filter without pseudointersection, which is a set almost included in each element of the filter. Finally, the ultrafilter number u is the minimal cardinality of a base of a free ultrafilter. It is well known that b ≤ d ≤ c, and p ≤ b ≤ u ≤ c, and that there are models for which p < b (see, for example, [5]).
We say that a family A is centered (has fip) if the intersection of any finite subfamily is nonempty; a family A is strongly centered (has sfip) if the intersection of any finite subfamily is infinite. If A and B are families of sets, then we say that A and B are compatible if A ∪ B is centered. If A = {A} we say that A is compatible with B.
If a family A is centered, then we denote by A the filter generated by A. Let A be an infinite subset of . A filter F on is said to be cofinite on A whenever U ∈ F if and only if A \ U is finite. A filter F is said to be cofinite if it is cofinite on some A ⊂ . It is well-known that a filter is free on A if and only if it includes the cofinite filter of A.

A relation between sets and functions.
Let W be a partition (1). For each n < , let w n k k< be an increasing sequence such that If F ∈ W, then, by definition, there exists the least n F < such that W n \ F is finite for each n ≥ n F . Now, for each n ≥ n F , there exists a minimal k n < such that w n k ∈ F for each k ≥ k n . Let ff F denote the set of those functions f for which Then E W (f, n F ) is the same for each f ∈ ff F , and it is the largest set of the form Conversely, for every function f ∈ , we define a family W f of subsets of as follows: F ∈ W f if there is n F < such that F = E W (fF , n F ). Therefore, we can state the following.
Proposition 2.2. The family f∈ W f is a base of W.

Quasi-subbases.
We say that a family A is finer than B if B ⊂ A . Moreover, A is called a quasi-subbase of (a filter) F if there exists a countable family C such that A ∪ C = F. Accordingly, A is quasi-finer than B if there exists a countable family C such that A ∪ C is centered and B ⊂ A ∪ C . Finally we say that a family is a P + -family if it is quasi-finer than no contour.
If W is a partition (1), then for each i < , let Proof. The implication 2 ⇒ 1 is evident. We will show 1 ⇒ 2. Suppose the contrary, and let B be a countable family of sets such that W ⊂ A ∪ B . Taking finite intersections i≤n B i instead of B n , we obtain a decreasing sequence so that, without loss of generality, we can assume that B = {B n } n< is decreasing. Since (2) is false, for each n there exists A n ∈ W such that A n ∈ A ∪ W ∪ {B n } .
Without loss of generality, for each n there is k(n) ≥ n such that A n ∩ W i is empty for all i < k(n) and Clearly C α does not include W for each α < . Thus, for each α < , there exists a set D α ∈ W such that D α ∈ C α . Let g α ∈ f f Dα for each α < . As < b, the family {g α } α< is bounded by some function g. Let G ∈ W g . We will show that G ∈ α< C α , hence G ∈ C α for each α < . Suppose not, and let α 0 be a witness. By construction, there exists n 0 < such that G ⊂ B c n 0 ∪ D α 0 . As (D α 0 ∪ B c n 0 ) ∩ B n 0 ⊂ D α 0 ∈ C α 0 , it follows that D α 0 ∪ B c n 0 ∈ C α 0 , and so G ∈ C α 0 . The anonymous referee noticed that Corollary 2.6 easily follows from [7, Proposition 2.28] by A. R. D. Mathias by argument pointed out in Mathias' proof. One can also prove Corollary 2.6 inductively using Theorem 2.5 and Proposition 2.1.
Corollary 2.6 [7] (b = c). If A is a strongly centered P + -family of subsets of , then there exists a P-point p such that A ⊂ p.
Theorem 2.7. Let u be an ultrafilter. If f(u) = RK u, then there exists U ∈ u such that f is one-to-one on U. §3. Applications: RK-ordering of P-points. M. E. Rudin [10] proved that, under CH, for each P-point p there exists a P-point q strictly RK-greater than p. Some years later, A. Blass [1,Theorem 6] proved this theorem under p = c.
The referee also noticed that Theorem 3.1 is easily derivable from [7, Proposition 2.28] by A. R. D. Mathias combined with [1, Theorem 6] by A. Blass. Nevertheless we present our original proof because its methods will be used in the sequel. Proof. Let f ∈ be a finite-to-one function such that lim sup n∈P card (f -1 (n)) = ∞ holds for all P ∈ p. We define a family A as follows: A ∈ A if and only if there exist i < and P ∈ p such that card (f -1 (n) \ A) < i for each n ∈ P. Then, Theorem 2.7 ensures that the ultrafilters we are building are strictly RK-greater than p.
We claim that f -1 (p) ∪ A is a P + -family. Suppose not, and take a witness W. From Remark 2.4, without loss of generality, we may assume that W ⊂ f -1 (p) ∪ A ∪ W . Consider two cases: Case 1: There exists a sequence (B n ) n< and a strictly increasing k ∈ such that B n ⊂ W k(n) , f(B n ) ∈ p, and B is compatible with f -1 (p) ∪ A ∪ W, where B = n< B n . Take a sequence (f( i≤n B n )) n< . This is an increasing sequence, and it is clear Note that R ∩ V i is finite for each i < , and that To complete the proof of the theorem use Corollary 2.6.
The following two, probably known, facts will be needed for Theorem 3.4 that extends, under CH, Theorem 3.1.

Fact 3.2. Let A be a centered family of subsets of such that A ∪ {F } is not an ultrafilter subbase for any F compatible with A. Let F be a countable family compatible with A. Then A ∪ F is not an ultrafilter subbase.
Proof. Without loss of generality, we may assume that (F n ) n< is a decreasing sequence of sets, such that F n+1 ∈ A ∪ {F n } . Put B n = F n \ F n+1 and define B 1 = n< B 2n , B 2 = n< B 2n+1 . Clearly at least one of sets B 1 , B 2 interact A-say B 1 does. If B 1 ∈ A ∪ F then we are done. Suppose that B 1 ∈ A ∪ F , and denote by n 0 , the minimal n < that B 1 ∈ A ∪ {F n } . But F n 0 +1 ∩ B 1 = F n 0 +2 ∩ B 1 and so F n 0 +2 ∈ A ∪ F n 0 +1 , which is a contradiction.  Proof. First repeat the proof of Theorem 3.1 except for the last line. Then continue as follows.
We arrange all contours in a sequence ( W α ) α<b and in a sequence (f ) <b . We will build a family {(F α ) α<b } <b of increasing b-sequences (F α ) α<b of filters such that: α+1 . The existence of such families follows by a standard induction with sub-inductions using Theorem 2.5 and Fact 3.3 for Condition 5. It follows from the proof of Fact 3.2 that F α is not an ultrafilter subbase for each α and . It suffices now to take for each < c, any ultrafilter extending <c F α and note that, by Proposition 2.1, it is a P-point.
A. Blass [1,Theorem 7] also proved that, under p = c, each RK-increasing sequence of P-points is upper bounded by a P-point. By Level n (T ) we denote level n in the tree T.
is an RK-increasing sequence of P-points, then there exists a P-point u such that u > RK p n for each n < .
Proof. For each n < we let f n to be a finite-to-one function that witnesses p n+1 > RK p n . Consider a sequence (T n ) n< of disjoint trees such that for each n < Since L ∞ = n< max T n is countably infinite we treat it as as well as L m = n< Level m (T n ). Let g m : L ∞ \ k<m L k → L m be a function defined by order of the trees T n .
On L ∞ we define a family of sets: B = n< g -1 m (p m ). To conclude it suffices, by Corollary 2.6, to show that B is a P + -family, thus by Theorem 2.5 it suffices to prove that g -1 m (p m ) is a P + -family, for any m. But this is an easier version of the fact which we established in the proof of Theorem 3.1.
In [1], A. Blass asked (Question 4) which ordinals could be embedded in the set of P-points, noticing that such an ordinal could not be greater than c + . The question was also considered by D. Raghavan and S. Shelah in [8] and answered, under MA, by B. Kuzeljević and D. Raghavan in a recent paper [6].
We prove that, under b = c, there is an order embedding of each ordinal less than c + into P-points. To this end, we need some (probably known) facts associated with the following definition: we say that a subset A of of nondecreasing functions such that f α (n) ≤ n for each n < and α < b.
Proof. First, we build, from the definition of b, an < * -increasing sparse sequence (g α ) α<b ⊂ of nondecreasing functions that fulfill the following condition: if α < < b, then g α (n) > g (2n) + n for almost all n < . Then a b-sequence (f α ) α<b defined by f α (m) = m -max {n : g α (n) < m} is as desired.   Proof. Facts 3.6 and 3.8 clearly imply that the first ordinal number which cannot be embedded as a sparse sequence in under id is equal to α or to α + 1 where α is a limit number. Facts 3.6 and 3.8 also imply that the set of ordinals less than α is closed under b sums.
Indeed, let be the minimal ordinal number < b + that may not be embedded under identity as an < * -increasing sparse sequence. Clearly cof ( ) ≤ b. Take an increasing sequence (α ) <cof that converges to . Clearly for each α < there is (g α ) <α -an embedding of α into as a sparse sequence under identity. By Fact 3.8 for each α < there is an < * -increasing sparse sequence of (f α ) <α such that f α < * f α < * f α+1 (for f α , f α+1 from the proof of Fact 3.6). Now (f α ) α<cof ( ), <α with lexicographic order is a required embedding of . Thus, this number is not less than b + . Proof. Note that cof ( ) ≤ b. Consider a set of pairwise disjoint trees T n such that each T n has a minimal element, each element of T n has exactly n immediate successors, and each branch has the highest .
Let {f α } α< ⊂ be a sparse, strictly < * -increasing sequence, the existence of which is demonstrated by Fact 3.9. For each α < , define Level fα (n) T n that agrees with the order of trees T n for n < such that f α (n) < f (n). Note that dom f α is cofinite on X for each α < .
Let p = p 0 be a P-point on X 0 = n< Level 0 T n . We proceed by recursively building a filter p on X . Suppose that p α are already defined for α < . If is a successor, then it suffice to repeat a proof of Theorem 3.1 for P -1 and f -1 .
So suppose that is limit. Let R ⊂ be cofinite with and of order type less than or equal to b. Define a family which is obviously strongly centered.
Clearly each filter that extends C is RK -greater than each p α for α < . But we need a P-point extension. Thus, by Corollary 2.6 it suffices to prove that C is a P + -family. Thus, by Theorem 2.5 it suffices to prove that ∈R, ≤α {(f ) -1 (p )} ⊂ (f α ) -1 (p α ) is a P + -family, for each α ∈ R. But it is (an easier version of) what we did in the proof of Theorem 3.1.
By Theorems 3.5 and 3.10 the following natural question arises: Question 3.11. What is the least ordinal α such that there exists an unbounded embedding of α into the set of P-points? 5 A. Blass [1,Theorem 8] also proved that, under p = c, there is an order-embedding of the real line into the set of P-points. We will prove the same fact, but under b = c. Our proof is based on the original idea of set X defined by A. Blass. Therefore, we quote the beginning of his proof, and then use our machinery.

Theorem 3.12 (b = c). There exists an order-embedding of the real line into the set of P-points.
Proof. --------(beginning of quotation) --------Let X be a set of all functions x : Q → such that x(r) = 0 for all but finitely many r ∈ Q; here Q is the set of rational numbers. As X is denumerable, we may identify it with via some bijection. For each ∈ R, we define h : X → X by The embedding of R into P-points will be defined by → D = h (D) for a certain ultrafilter D on X. If < , then We wish to choose D in such a way that (a) D ∼ = D (therefore, D < D when < ), and (b) D is a P-point.
Observe that it will be sufficient to choose D such that (a') D ∼ = D when < and both and are rational, and Condition (a') means that, for all < ∈ Q and all g : X → X , D = g(D ) = gh (D ). By Theorem 2.7, this is equivalent to {x : gh (x) = x} ∈ D , or by our definition of D , (a") We now proceed to construct a P-point D satisfying (a") for all < ∈ Q and for all g : X → X ; this will suffice to establish the theorem.
We claim that A is a P + -family. Indeed, by Theorem 2.5, it suffices to prove that for each n < , i<n A i is a P + -family. Suppose not and take (by Remark 2.4) the witnesses i 0 and W such that W ⊂ i<i 0 A i ∪ W . For each n < , consider the condition (S n ): ) > n . Case 1: S n is fulfilled for all n < . Then, for each n < , j < n, choose x j n ∈ W n such that h i (x j n ) = x n and h i (x Clearly E c ∈ W, but E ⊂ i<i 0 g∈G (A g, i , i ) ∪ l<m W l for any choice of finite family G ⊂ X X and for any m < .
Case 2: S n is not fulfilled for some n 0 < . Then, there exist functions {g n,i } n≤n 0 ,i<i 0 ⊂ X X such that W 1 ⊂ n≤n 0 i<i 0 (A g n,i , i , i ) ∪ n≤n 0 W n , i.e., W¬ is not compatible with i<i 0 A i ∪ W . Corollary 2.6 completes the proof. The long line is defined as L = 1 × (0, 1] ordered lexicographically. If f : Y → , then the support of f is defined as Proof. We will combine ideas form proofs of Theorems 3.1 and 3.12 with some new arguments. Again, let X be a set of all functions x : Q → such that x(r) = 0 for all but finitely many r ∈ Q. Since X is infinitely countable, we treat it as .
Let p be a P-point on X such that for each q ∈ Q and for each P ∈ p there exists x ∈ P such that max supp (x) < q. Let f ∈ X X be a finite-to-one function such that lim sup x∈P card (f -1 (x)) = ∞ for all P ∈ p and that max supp x < max supp f(x). Again, we define a family A as follows: A ∈ A if and only if there exist i < and P ∈ p such that card (f -1 (x) \ A) ≤ i for each x ∈ P. For each ∈ R, we again define functions h : X → X by

List all rational numbers in -sequence
Our aim is to prove that B can be extended to such a P-point Q that h (Q) = h (Q) for each = ∈ Q (and thus for each = ∈ R).
(4) To this end, we add to B a family C defined as follows: list all pairs ( , ), < ∈ Q in the sequence ( i , i ) i< . For each g ∈ X X , < ∈ Q, define C g, , = {x ∈ X : Thus to prove (4), it suffices by Corollary 2.6 to prove that B ∪ C is a P + -family. Thus, by Theorem 2.5, in order to prove (4), it suffices to prove that: D i is a P + -family for D i defined for i < as follows: First, to prove it, we notice that D i is strongly centered. Indeed, define Fix i, and suppose that (5) does not hold. So take a witness W. From Remark 2.4, without loss of generality, we may assume that W ⊂ D i ∪ W .
Let A W ∈ A, P W ∈ p, n W ∈ , C W n ∈ C for n ≤ n W , and l W ∈ . Define Define W ∈ Wif and only if W ∈ W and W is co-finite or empty on each W i . We will say that a set W ∈ W is attainable (by (n A , P W , n W , l W )) if there exist A W ∈ A n A , {C W k ∈ j≤i C j : k ≤ n W } such that the condition W * (A W , P W , C W 1 , ... , C W n W , W l W ) ⊂ W is satisfied. The complement (to W 1 ) of the attainable set is called removable and sometimes we indicate which variables, sets, functions.
Since W ⊂ D i , thus each set W ∈ Wis attainable. Consider a sequence of possibilities: 1) l cannot be fixed, i.e., for each l ∈ there exists W ∈ Wsuch that W is not removable by any (n A , P W , n W , l); 2) l can be fixed, but n A cannot, i.e., for each W ∈ W -, W is attainable by some (n A W , P W , n C W , l), but for each n A there exists W ∈ Wsuch that W is not attainable by any (n A , P W , n C W , l); 3) l and n A can be fixed, but n C cannot; 4) l, n A and n C can be fixed, but P cannot; 5) l, n A , n C , and P can be fixed. Note that each set W ∈ Wis attainable if and only if an alternative of cases 1) to 5) holds.
In case 1) for each l, let W l ⊂ W l and W l ∈ Wbe a witness that l may not be fixed. Note that l< W l ∈ Wand that l< W l may not be removed by any (n A W , P W , n C W , l W ). In case 2) we proceed like in case 1). Note that if l ≥ l and n A and a set W ∈ Wis not removable by any (n A , P W , n C W , l ) then the set W is also not removable by any (n A , P W , n C W , l). Thus it suffices to consider cases when l = n A . For each l, let W l ⊂ W l and W l ∈ Wbe a witness that l and n A = l may not be fixed. Again note that l< W l ∈ Wand that l< W l may not be removed by any (n A W , P W , n C W , l W ). In case 3) we proceed just like in case 2), not using that n A is fixed. In case 4) for k < , let X k be the set of those x ∈ X that for all U ⊂ f -1 (x) such that card (f -1 (x) \ U ) ≤ n A , for all partitions of a set i j=1 ( -1 j (X ) ∩ (W k )) on the sets X m,n , for m < n ≤ i, there exist m 0 , n 0 such that m 0 < n 0 ≤ i and there exist x 1 , ... , x n C +1 ∈ X m 0 ,n 0 that for (min) = min { m 0 , n 0 }, (max) = max { m 0 , n 0 } there is (min) (x r ) = (min) (x j ) for r, j ≤ n C + 1 and (max) (x r ) = (max) (x j ) for r, j ≤ n C + 1, r = j.
Clearly, (X k ) is a decreasing sequence. If there exists k such that X k ∈ p then putting l = k there exists a set P = (X k ) c such that all W ∈ Wmay be attained by (n A , P, n C , l) so we would be in case 5), so, without loss of generality, X k ∈ P for each k < .
Thus take a partition of X by (X k \ X k+1 ). Since p is a P-point, and since X k ∈ p thus there exists P 0 ∈ p such that P k = P 0 ∩ (X k \ X k+1 ) is finite for all k < .
For each x ∈ P k there exists a finite(!) set K k,x ⊂ (W k ) ∩ j≤i h -1 i that may not be removed by (n A , X, n C , k). (The proof that K k,x may be chosen finite is analogical, but easier, to that of case 5)). Take K = k< ,x∈P k K k,x and notice that (W 1 ) \ K ∈ Wand that (W 1 ) \ K is not removable by (n A , P, n C , l) for any P ∈ p.
In case 5) arrange j≤i h -1 j (P) ∩ W l into a sequence (x k ) k< . Let R(x) = card (f -1 (x)) card (f -1 (x))-n C , where n k denotes a binomial coefficient, and let (A x,r ) r≤R be a sequence of all subsets of f -1 (x) of cardinality equal to card (f -1 (x)n C ). Consider a tree T, where the root is ∅ and on a level k the nodes are pairs of natural numbers j, r such that j ≤ i and r ≤ R(f(x k )) and, for each branchT of T, 2 (T (k 1 )) = 2 (T (k 2 )) if f(x k 1 ) = f(x k 2 ), whereT (k) is an element of level k of a branchT and 2 is a projection on the second coordinate. We see j as a choice to which class C j does a set C (.) (.) belongs and we see r as a choice of one of sets A f(x),r that C (.) (.) together with A f(x),r removes x k . Clearly, the complements of all finite sets belong to W, so each finite set is removable.
(6) The maximal element of the branchT has no successors if and only if there is j ≤ i such that there is no n sets in C j that remove all x t such that T (k) = j and f(x k ) ∈ A f(x k ), 2 (T (k)) . It implies that the set {x k :T (k) = j, f(x k ) ∈ A f(x k ), 2 (T (k)) } contains more than n different elements, say x 1 , ... , x n+1 , such that h j (x s 1 ) = h j (x s 2 ) and h j (x s 1 ) = h j (x s 2 ) for s 1 = s 2 , s 1 , s 2 ∈ {1, ... , n + 1}.
By König Lemma if all branches are finite, then the height of the tree T is finite, and so there are irremovable finite sets in contrary to (6). Thus there is infinite branch and the whole set j≤i h -1 i (P) ∩ W l is removable. As an immediate consequence of Lemma 3.13 (with the use of Theorem 3.5) we have the following: Remark 3.15. Note that there is a potential chance to improve Theorem 3.14 in the virtue of Question 3.11, i.e., if, in some model, for each α < κ (for some cardinal invariant κ) each RK-increasing α -sequence of P-points is upper bounded by a P-point, then (in that model) if b = c, then, above each P-point, there is an order embedding of a κ-long-line into the set of P-points. §4. Cardinal q. An inspection of our proofs indicates a possibility of refinement of most results with the aid of an, a priori, new cardinal invariant. We define q to be the minimal cardinality of families B, for which there exists a family A such that A ∪ B includes a contour, and A ∪ C includes no contour for every countable family C. 6 If P is a collection of families such that P ∈ P whenever P includes a contour, then q fulfills Each contour has a base of cardinality d, which, by the way, is the minimal cardinality of bases of contours [13, Theorem 5.2]. Therefore, taking into account Theorem 2.5, we have Using the cardinal q, we are in a position to formulate stronger versions, if b < q is consistent, of several of our theorems with almost unchanged proofs. Indeed, by the proof of Theorem 3.1 we get the following theorem: Theorem 4.2 (q = c). For each P-point p there exists a P-point q strictly RKgreater than p.
By the proof of Theorem 3.5, we have Theorem 4.3 (q = c). If (p n ) n< is an RK-increasing sequence of P-points, then there exists a P-point u such that u > RK p n for each n < .
By the proof of Theorem 3.10, we get Theorem 4.4 (q = c). For each P-point p, for each α < b + , there exists an order embedding of α into P-points above p.
By the proof of Theorem 3.12, we obtain By the proof of Theorem 3.14, we have A relative importance of the facts formulated above depends on answers to the following quest.
Question 4.7. Is q equal to any already defined cardinal invariant? Is b < q consistent? Is q < d consistent? §5. Variants of invariants. The cardinal q can be seen as an instance of cardinal invariants, which can possibly be defined in order to refine certain types of theorems, by scrutinizing the mechanisms underlying their proofs. In our approach, such cardinals represent "distances" between certain classes of objects. They carry some obvious questions about their relation to the usual cardinal invariants, and in particular to those that they are supposed to replace in potentially refined arguments.
Let S and T be collections of families (of sets or functions, or possibly other objects) such that for each S ∈ S there exists T ∈ T such that S ⊂ T . For each S ∈ S, we define Moreover, if α is a limit ordinal, and the cardinals dist (S, T) are defined for all < α, then dist <α (S, T) = sup <α dist (S, T).
In particular, if S denotes the collection of compatible with a contour families of subsets of , and T stands for the collection of families including a subbases of a contour, then we write q α = dist α (S, T). In order to show that (qα) α are variants of q, we need the following Alternative Theorem [4, Theorem 3.1]. A relation A ⊂ {(n, k) : n < , k < } is called transversal if A is infinite, and {l : (n, l ) ∈ A} and {m : (m, k) ∈ A} are at most singletons for each n, k < . If F is compatible with G, then the following alternative holds : F n is compatible with G k for a transversal set of (n, k), or F is compatible with G k for infinitely many k, or F n is compatible with G for infinitely many n.
Proof. By taking S ∈ T and B = ∅, we infer that q 0 = 0. To see that q 1 = 1, let A, B be disjoint countably infinite sets. Let S A be a contour on A and let S B be a cofinite filter on B. Define a filter S on A ∪ B by S ∈ S if and only if S ∩ A ∈ S A and S ∩ B ∈ S B . Clearly S is not finer than a contour (since is RK-smaller than a cofinite filter S B ), and S ∪ {A} is a subbase of a contour.
Clearly, q 2 cannot be finite. To see that q 2 = ℵ 0 , let W = (W n ) n< be a partition of into infinite sets. We define a family S so that S ∈ S if and only if S is cofinite on each W n . Suppose that there is a partition V = (V n ) n< such that V ⊂ S ∪ S 0 for some set S 0 such that S ∪ S 0 is strongly centered. Let N fin = {n < : card (W n ∩ S 0 ) < } and N ∞ = \ N fin . Define S fin = S 0 ∩ n∈N fin W n and S ∞ = S 0 ∩ n∈N∞ W n . Since S c fin ∈ S, without loss of generality, we can assume that S 0 = S ∞ and so without loss of generality we can assume that S 0 = .
Note also that n< V n ∩ W i is infinite for infinitely many i, and so W is compatible with V. Thus we meet the assumptions of Theorem 5.1, and in each of the three cases there exist i, j < such that V i ∩ W j is infinite. But \ V i ∈ V and thus \ (V i ∩ W j ) ∈ S, contrary to the definition of S. On the other hand, by adding W to S, we obtain a subbase of W.
That q 3 = q follows directly from the definition of q. Finally, q α ≤ d since each contour has a base of cardinality d, as we have shown in [13, Theorem 5.2].
If S is the collection of strongly centered families, but T is the collection of free ultrafilter subbases, then, taking S as an empty family, clearly dist(S, T) = u and so dist α (S, T) = u for some α, thus we obtain variants of u. By Fact 3.2, Fact 5.3. u 0 = 0, u 1 = 1, u 2 ≥ ℵ 1 .
By the proof of Theorem 3.4, we obtain: Theorem 5.4 (q = u 2 = c). If p is a P-point, then there is a set U of Rudin-Keisler incomparable P-points such that card U = c and u > RK p for each u ∈ U. A similar approach can be carried out for all other cardinal invariants. Its usefulness, however, depends on the way these cardinals are used in specific arguments.