HIGH DIMENSIONAL SEQUENTIAL COMPACTNESS

. We give examples of n -sequentially compact spaces that are not ( n + 1) -sequentially compact under several assumptions. We improve results from Kubis and Szeptycki by building such examples from b = c and ♦ ( b )+ d = ω 1 . We also introduce a new splitting-like cardinal invariant and then show that the same holds under s = b .


Introduction
A widely studied topological property is that of sequential compactness.Recall that a space X is sequentially compact if every sequence {x n : n ∈ ω} ⊆ X has a convergent subsequence.
An important fact about this property is that it characterizes compactness in metric spaces, that is, a metric space is sequentially compact if and only if it is compact.Nevertheless, this fact is not true in general, since ω 1 is a sequentially compact non compact space and βω, the Stone-Cěch compactification of a countable discrete space, is compact but fails to be sequentially compact (see [54] ).
In [37], Kubiś and Szeptycki introduced higher dimensional versions of sequential compactness and showed that these properties also hold in metric spaces.The 2dimensional case had been previously introduced and proved to hold in compact metric spaces in [6] by Bojańczyk, Kopczyński and Toruńczyk.There the authors used this to obtain idempotents in compact semigroups.
We follow the convention that [B] n denotes the family of all subsets of B of cardinality n.Analogously, [B] ω denotes the family of all of its countable infinite subsets and [B] <ω = n∈ω [B] n .Recall that if X is a topological space, p ∈ X, B is a countable set and f : B → X, then we say that f converges to p if for all U ⊆ X open such that p ∈ U , there exists a finite set F ⊆ B such that f ′′ (B \ F ) ⊆ U .It is natural to generalize this definition to higher dimensions.Following [37], if n ≥ 1 and f : [B] n → X, then f converges to p if for every neighborhood U of p there is a finite set F ⊆ B such that f ′′ [B \ F ] n ⊆ U and in this case we say that the function is f convergent 1 .Definition 1. [37] Let X be a topological space and n ≥ 1.We say that X is an nsequentially compact space if for every function f : [ω] n → X there exists B ∈ [ω] ω such that f ↾ [B] n is convergent.We say that f ↾ [B] n is a convergent subsequence of f .
Observe that being 1-sequentially compact is equivalent to being sequentially compact.It is also not hard to see that Ramsey's Theorem2 is equivalent to the assertion that every finite space X is n-sequentially compact for all n ≥ 1.
A similar notion was studied by Helmut Knaust in [35] and [36], where the author defined Ramsey-uniformly convergence and making use of it, he defined a space to have the Ramsey property if any double sequence {x(i, j) : i, j ∈ ω} such that lim i→∞ lim j→∞ = x also converges Ramsey-uniformly to x.The notion of Ramseyuniformly convergence coincides with the notion of convergence introduced in [37] in the 2-dimensional case.It was shown by Knaust that any pointwise compact subset of the Baire-1 functions on a Polish space has the Ramsey property [36], that any angelic space so that χ(x, A) < p for all countable relatively compact subsets A in X and all x ∈ X has the Ramsey property and that so do all spaces of continuous functions C(Y ) with the topology of pointwise convergence where Y is a quasi Suslin space [35].This result in the particular case of first countable spaces (and then for the real numbers with their usual topology) was previously showed by Rosenthal (see [48]).
In [37], the term n-Ramsey was used instead of n-sequentially compact, however, it seems that the latter is more accurate in this context as the term n-Ramsey conflicts with other previously introduced notions and does not stress the compactness nature of the property.Note that the Ramsey property and being 2-Ramsey (what we are calling now 2-sequentially compact) are similar but different.While the Ramsey property ensures a Ramsey-like convergence of a double sequence that already comes with a limit point, 2-sequential compactness also claims the existence of such a point x ∈ X for every function f : [ω] 2 → X.For this reason and to avoid confusion with Knaust's property and Ramsey spaces introduced by Todorčević [51], we have decided to adopt the more descriptive name of n-sequential compactness In [37], Kubiś and Szeptycki proved the following result: Theorem 2. [37] Every compact metric space is n-sequentially compact for every n ≥ 1.
Thus we can not differentiate the class of n-sequentially compact spaces from the class of m-sequentially compact spaces with n = m in the realm of metric spaces.So, the interesting question is whether we can distinguish these properties in the general category of topological spaces.
It is easy to see that every (n+1)-sequentially compact space is also n-sequentially compact (Proposition 2.2 in [37]) and Kubiś and Szeptycki proved the following: Theorem 3. [37] (1) There is a compact sequentially compact space (that is, 1-sequentially compact) that is not 2-sequentially compact.(2) Under CH, for every n ∈ ω, there exists a n-sequentially compact space that is not (n + 1)-sequentially compact.
In this work we will improve item (2) of Theorem 3 by constructing n-sequentially compact spaces that are not (n + 1)-sequentially compact under several cardinal invariant assumptions.We also show that there is Fréchet example for item (1) answering a question posed by the authors in [37].This example was sketched in that paper under the assumption of a completely separable mad family, which follows under, e.g., c ≤ ℵ ω .We point out that the construction does not need a maximal almost disjoint family and since the existence of completely separable almost disjoint families follows from ZFC, an easy modification of the result in [37] can be carried out without extra assumptions.Although we will be quoting theorems from [37], it is not needed that the reader is familiar with it, since we will be recalling the needed notions and results.
The theory of n-sequentially compact spaces is just in its beginnings and we expect it to be richer and more interesting than the one of sequentially compact spaces.This is our humble contribution to this new theory.

Preliminaries
All spaces considered here are Hausdorff.Denote by ω the first infinite ordinal, by ω 1 the first uncountable ordinal and by c the cardinality of R. Given two infinite sets A, B ⊆ ω, we write A ⊆ * B if A \ B is finite.Two sets A, B are almost disjoint if A ∩ B is finite.A family A ⊆ [ω] ω is almost disjoint (ad) if its elements are pairwise almost disjoint and it is maximal almost disjoint (mad) if it is ad and is not properly contained in any other almost disjoint family.
We say that a set ω is a splitting family if for every X ∈ ω there exists S ∈ S such that S splits X.
The set of functions from X to Y will be denoted by X Y .For two functions f, g ∈ ω ω , we write f ≤ * g if {n ∈ ω : f (n) > g(n)} is finite.A family B ⊆ ω ω is unbounded if for every f ∈ ω ω there exists b ∈ B such that b * f and B is dominating if for every f ∈ ω ω there exists d ∈ B such that f ≤ * d.We also say that B is a scale if B is a dominating family and it is well ordered by ≤ * .
We define below the cardinal invariants used in this work for the ease of the reader, • a is the minimum size of a mad family, • s is the minimum size of a splitting family, • b is the minimum size of an unbounded family, • d is the minimum size of a dominating family.
Each of these cardinals is uncountable and smaller or equal to c.It is also well known that b ≤ d, b ≤ a and s ≤ d and there are no other provable relations among them in ZFC.It is worth noting that scales exist if and only if b = d.A nice survey about cardinal characteristics and their properties is [5].
A family I ⊆ P(ω) is an ideal if it is closed under finite unions and subsets, i.e., if A, B ∈ I then A ∪ B ∈ I and if A ⊆ B and B ∈ I then A ∈ I.An ideal I is tall if for every X ∈ [ω] ω there exists I ∈ I such that |X ∩ I| = ω.Given a tall ideal I, the cardinal cov * (I) is defined as follows: We can also define cof(I), the cofinality of an ideal I, as the minimum size of a subfamily of I that generates it, that is: For more on ideals on countable sets and their cardinal invariants the reader may consult [30], [26], [32], [44] and [45].Now, we summarize some definition and results of almost disjoint families that will be helpful in our constructions.Definition 4. If A is an almost disjoint family, we denote the ideal generated by A as I(A).We always assume that our ideals contain the finite sets, then Here, B denotes the ideal generated by B. Clearly an element of I(A) is a set that can be almost covered by a finite subfamily of A.
Definition 5.For an almost disjoint family A we define Note that I ++ (A) ⊆ I + (A) and A is mad if and only if, I ++ (A) = I + (A).If A is an almost disjoint family on a countable set N , then its Isbell-Mrówka space, Ψ(A), is N ∪ A, where N is the set of isolated points and the basic neighborhoods of each A ∈ A are of the form {A} ∪ A \ F for F ∈ [N ] <ω .The space Ψ(A) has interesting properties, it is Hausdorff, zero-dimensional, separable and locally compact.The Franklin space of A is then the one point compactification of Ψ(A), i.e., F (A) = Ψ(A) ∪ {∞}.The basic neighborhoods of ∞ are of the form: where n ∈ ω , A 0 , . . ., A n ∈ A and F ∈ [N ] <ω .To learn more on Isbell-Mrówka spaces the reader may consult the survey [24] or the articles [25], [31], [47], [18], [16] and [9].
Since their early beginnings the Mrówka-Isbell spaces Ψ(A) have turned out to be a fruitful source of examples and counterexamples in topology, since combinatorial properties of A translate in topological properties of Ψ(A).Plenty of applications of Mrowka-Isbell spaces to topology and functional analysis can be found in the literature.However, constructing mad families with special combinatorial properties is usually a difficult task without extra assumptions beyond ZFC.An example of ad families with special combinatorial properties are the completely separable ones.Definition 6.An almost disjoint family A is completely separable if for all X ∈ I ++ (A) there exists A ∈ A such that A ⊆ X.
These families were introduced by Hechler in [23] and in [11].Erdős and Shelah asked if there exist completely separable families that are mad.Most of the early work on this question was done by Balcar and Simon in [2], who proved that completely separable mad families exist under many assumptions.The next major advance on this was done by Shelah in [49], who introduced a technique that allowed him to prove that there are completely separable families under s < a and also under s ≥ a + a certain "PCF hypothesis".The last big progress in the search of these families was done by Mildenberger, Raghavan and Steprāns in [40] and [46] who improved and extended the Shelah's technique in order to get the existence of a completely separable mad family under s ≤ a.This technique has turned out to be very powerful not only in the construction of completely separable mad families but also for other special mad families, e.g, weakly tight mad families (under s ≤ b) and +-Ramsey mad families (see [46] and [20] respectively).This technique will play a crucial role in Section 8.The interested reader may consult [27] and [19].
Finally, Simon proved the following remarkable theorem: Theorem 7. [15] There is a completely separable nowhere mad family.
So we can find a completely separable almost disjoint family in ZFC if we do not demand maximality on it.As we will see in the Section 4, this family will lead to a sequentially compact space not 2-sequentially compact.The recursive construction can be carried on thanks to the following result: Lemma 8. See [15] and [19].If A is a completely separable almost disjoint family and X ∈ I ++ (A), then the set {A ∈ A | A ⊆ X} has cardinality c.
Recall that a space X is Fréchet if whenever x ∈ A, with A ⊆ X, there exists {x n : n ∈ ω} ⊆ A that converges to x. Obviously, every first countable space is Fréchet, but there are many Fréchet spaces that are not first countable.An almost disjoint family A is nowhere mad if for every X ∈ I + (A) there exists B ∈ [X] ω that is almost disjoint with every element in A.

Theorem 9. [27]
Let A be an almost disjoint family, then A is nowhere mad if and only if F (A) is a Fréchet space.
We now remind some important results concerning the n-sequential compactness property proved in [37].Proposition 10. [37] The Franklin space of a mad family is not 2-sequentially compact.
Let A be an almost disjoint family on a countable set N , f : [ω] n → F (A) and suppose we want to see if we can find a convergent subsequence of f .By passing to an infinite subset of ω if needed (this can be done by applying Ramsey's theorem), we may assume that one of the following cases holds: The following lemma tells us that we only need to worry in the last case: Lemma 11. [37] If A is an almost disjoint family on a countable set N of isolated points and n ≥ 1, F (A) is n-sequentially compact if and only if for all f : Recall that the character of x ∈ X is the minimum cardinality of a local base of x and it is denoted by χ(x, X).The character of X is χ(X) = sup{χ(x, X) : x ∈ X}.Theorem 12. [37] Every sequentially compact space with character less than b is n-sequentially compact for all n ≥ 1.
In Theorem 57 we will see that b is optimal for the Theorem above.A consequence of Theorem 12 and Lemma 11 is: Corollary 13. [37] Let A be an almost disjoint family on a countable set N , such that for every f : [ω] n → N there exists X ∈ [ω] ω so that f ′′ [X] n has closure of size less than b in Ψ(A).Then F (A) is n-sequentially compact.
We also get: Corollary 14. [37] Let A be an almost disjoint family on a countable set N of isolated points and n ≥ 1.If for all f : [ω] n → N , there exists B ∈ [ω] ω such that One of the key features of the counterexamples given in [37] is the structure of the FIN n ideal on ω n .This ideals were introduced by Katetov in [34].Recall that FIN = [ω] <ω denotes the ideal of finite sets of ω.The ideal FIN n is defined recursively as the Fubini product of To learn more about this ideals, the reader may consult [3], [22], [38] and [39].The following combinatorial lemma will be very useful for us in the future.Lemma 15. [37] For every f : To deal with these kind of functions, we will also need an useful tool known as the canonical Ramsey theorem.A very deep and powerful fact due to Erdös and Rado is that there is a finite, very easy to describe "base" for the equivalence relations on [ω] n .
For the rest of this paper, given a ∈ [ω] n , we write it as {a i : i < n} where a 0 < • • • < a n−1 .If I ⊆ n, let a ↾ I := {a i : i ∈ I}.For every I ⊆ n, we can define the equivalence relation E I on [ω] n given as follows: Note that: Of course not every equivalence relation is of this form.However, the surprising fact is that every equivalence relation is "locally" one of the ones we just described.Formally, we have the following: The reader can learn more in this topic from Todorcevic's book [51].Applying Theorem 16 several times, we get the following: Corollary 17.Let E 1 , . . ., E l equivalence relations on [ω] n .There exist M ∈ [ω] ω and I 1 , . . ., I l ⊆ n such that for all a, b ∈ [M ] n and i ≤ l, we have that

Examples of n-sequentially compact spaces
In this section we will give some examples of n-sequentially compact spaces.By Theorem 12, we know that there are sequentially compact spaces that fail to even satisfy the 2-sequential compactness property and that, in general, the properties of n-sequential compactness and (n + 1)-sequential compactness may be consistently different.To see how subtle is to stratify the classes of n-sequentially compact spaces, we will analyze some classical examples of sequentially compact non first countable spaces.These are necessary assumptions due to the following results: Proposition 18. [37] (1) If n ≥ 1 and X is (n + 1)-sequentially compact, then X is n-sequentially compact.In particular if X is n-sequentially compact for some n ≥ 1, then X is sequentially compact.(2) If X is sequentially compact and first countable, then X is n-sequentially compact for all n ≥ 1.
The following five examples are all the sequentially compact, non first countable spaces that appear in the book [50] (which can also be conveniently accessed in the website [43]): (1) The Fort space.
We will describe them in detail below and see that indeed all of them are nsequentially compact for every n ≥ 1.Note that as an easy consequence of Proposition 18 item (2) we have that: Corollary 19.[37] Let X be sequentially compact space in which the closure of every countable set is first countable.Then X is n-sequentially compact for all n ≥ 1.
From the previous corollary, we have that the one-point compactification of every discrete space is n-sequentially compact for all n ≥ 1, in particular, the Fort space, that is, the one-point compactification of the discrete space of size c, is nsequentially compact for all n ≥ 1.
Example 20. (Closed long ray) Consider L = ω 1 × [0, 1) with the lexicographic order and L = L ∪ {∞} where ∞ is the maximum point.Then the closed long ray L with the order topology is n-sequentially compact for all n ≥ 1.
Proof.Let f : [ω] n → L, applying Ramsey's theorem we know that there exist In later case we are done and in the former note that f ′′ [B] n ⊆ L and L is first countable and sequentially compact so we can apply Corollary 19.
Note that L in Example 20 is sequentially compact but it is not first countable, since ∞ ∈ L has no countable base.
Proof.Let f : [ω] n → L, applying Ramsey's theorem we know that there exist But note that the topology of L as a subspace of L coincides with the topology of L as a subspace of L.
Let us denote by ∆ the set of points of the form The following example illustrates the usual way in which we will be applying Theorem 16 to derive n-sequential compactness of certain spaces.
with the following topology: if p = (x, x) ∈ ∆ then a basic neighborhood of p is of the form: n has a convergent subsequence 3 .Applying Theorem 16 to the function f 0 , we can get M ∈ [ω] ω and I ⊆ n such that for all s, t ∈ If I = ∅ then we are done by (i).So assume that I = ∅ and let us consider Finally, we will prove that in the realm of limit ordinals, being sequentially compact, having uncountable cofinality and being n-sequentially compact for all n ≥ 1 are equivalent.We will first need the following: As Y is a set of ordinals, by going to a subset if necessary, we can enumerate Y in order type ω Note that by the last proposition, if γ is a limit ordinal with uncountable cofinality and X ⊆ γ is countable, then the closure of X in γ is sequentially compact and first countable, so by Corollary 19 we have that: Corollary 24.Let γ be a limit ordinal.Then cof (γ) > ω if and only if γ with the order topology is n-sequentially compact for all n ≥ 1.
Finally, we get a full characterization of those ordinals that are n-sequentially compact for all n ∈ ω.
Proof.Let f : [ω] n → γ and assume that γ has uncountable cofinality.Let X = f ′′ [ω] n and define α = sup(X).Since γ does not have countable cofinality, α ∈ γ.It follows then that α = max(X).Take X with the inherit order, since the maximum of X exists, β = ot(X) is a countable successor ordinal.Therefore it is first countable and sequentially compact, which implies that β is n-sequentially compact by Proposition 18 item (2).In consequence, f has a convergent subsequence in X ⊆ γ.
With this we now know that all sequentially compact examples in [50] (and only those) are n-sequentially compact for all n ≥ 1.

Sequentially compact Fréchet spaces that are not 2-sequentially compact
It is well know that in the case of Franklin spaces sequentially compact does not imply being Fréchet, however, Paul Szeptycki pointed out that a 2-sequentially compact Franklin space must be Fréchet.
Proof.Suppose that F (A) is not Fréchet, this means, by Theorem 9, that A is somewhere mad, i.e., there exists In [37], the authors asked whether there is a ZFC example of a Fréchet sequentially compact space that is not 2-sequentially compact.They showed that a completely separable mad family suffices for the construction of such example.As discussed earlier, the existence of a completely separable mad family holds under very weak assumptions like c < ℵ ω , however, as we stated in Theorem 7, completely separable almost disjoint families, possibly not maximal, do exist in ZFC.The existence of one of these families in ZFC is good enough for the construction sketched in [37], we give a full proof below. If Proof.We will show that f ↾ [B] n does not converge to any point in The only thing that remains to be seen is that f ↾ [B] n does not converge to ∞.Let A ∈ A \ A 0 given by item (v).We will show that This shows that F (A) is not n-sequentially compact as witnessed by f .
We can now prove the following, answering a question in [37].
Theorem 28.There exists (in ZFC) a space which is Fréchet, sequentially compact and it is not 2-sequentially compact.
Proof.Let A be an almost disjoint family on ω × ω which is completely separable, infinite and nowhere mad.We can assume that given by G({m, n}) = (m, n), where m < n.We will show that A, A 0 = {A n | n ∈ ω} and f = G satisfy the hypothesis of Lemma 27 for n = 2. Conditions (ii), (iii) and (iv) are clear.For (v) note that (iii) and (iv) imply that Since A is completely separable, using Lemma 8, there exists A ∈ A \ A 0 as required in (v).The relevance of the cardinals b and d in our examples is evident due to the next few facts regarding cov * (FIN n ) and cof(FIN n ).We will first introduce the concept of Hechler trees in order to make the analysis of these results cleaner.
Given n ∈ ω, we will say that T ⊆ ω ≤n+1 is a Hechler tree, if for every s ∈ T ∩ ω ≤n , the set of successors of s is coinitial on ω, i.e., there exists k s ∈ ω such that succ T (s) = {i ∈ ω | s i ∈ T } = ω \ k s .We can naturally associate a function h T : T ∩ ω ≤n → ω to each Hechler tree T ⊆ ω ≤n+1 by defining h(s) = k s .
For s, t ∈ ω ≤n , we say that s ≺ t if either s ❁ t or s = r l, t = r m for some r ∈ ω ≤n and l < m, where s ❁ t means that s is a proper initial segment of t.Enumerate ω ≤n = {s i | i ∈ ω} such that s i ≺ s j implies i < j.We fix this enumeration once and for all.Definition 29.Let f ∈ ω ω and let T ⊆ ω n+1 be a Hechler tree.We define Let C s = {t ∈ ω n+1 | s ⊑ t} for every s ∈ ω ≤n and let us write C k instead of C (k) .Also, for a tree T ⊆ ω ≤n we denote its set of branches by [T ] = T ∩ ω n .It follows directly from the definition that for every X ∈ FIN n there exists a Hechler tree T ⊆ ω ≤n such that X ∩ [T ] = ∅.
Proof.For every X ∈ H let T X be a Hechler tree disjoint from X.We write f X instead of f TX .
Since |H| < b, there is a function f ∈ ω ω such that f > * f X for all X ∈ H.We will recursively define a block sequence (a m | m ∈ ω) ⊆ ω n+1 .Let (a i m : i ≤ n) denote the sequence a m .Assume we have constructed a k for every k < m.For a m define a 0 m ∈ B \ (f (∅) ∪ max(a m−1 )) (where a −1 = 0).Now for every i < n pick a i+1 m ∈ B \ f ((a 0 m , . . ., a i m )) bigger that a i m .These choices are possible since B is infinite.This finishes the To see that it works fix } for all but finitely many m ∈ ω.Given any a 0 m as above, a 0 m > f X (∅) and as (a 0 m , . . ., a i m ) extends a 0 m in ω n+1 for any i ≤ n, it follows from the assumption over m < k}, with the latter set being finite.The previous lemma gives in particular that b ≤ cov * (FIN n+1 ) for every n ≥ 1.The other inequality can also be proved using similar ideas.
Proof.We only need to prove that cov * (FIN n+1 ) ≤ b.Suppose that κ < cov * (FIN n+1 ) and let {h β ∈ ω ω | β ∈ κ}.Recall that b is the minimum size of a family B ⊆ ω ω that is unbounded on each infinite subset of ω (see [52]), so it is enough to see such that t l = r i and t l ′ = r j and i < j then l < l ′ .
For each β ∈ κ consider the set Now define f : ω → ω as follows: Now we are ready to prove that under b = c we can distinguish the classes of n-sequentially compact and (n + 1)-sequentially compact spaces for every n ≥ 1.
Theorem 32.(b = c) For all n ≥ 2 there exists a n-sequentially compact space which is not (n + 1)-sequentially compact.
Proof.We will construct an almost disjoint family A on ω n+1 such that its Franklin space is n-sequentially compact but not (n + 1)-sequentially compact.Let G : ) and {X α | ω ≤ α < c} ⊆ P(ω) such that the following hold: (1) For all n ∈ ω, A n = C n and A α is a block sequence for every α ≥ ω (hence Assume that we have already constructed {A β | β < α} and {X β | β < α} for some α ≥ ω.We can apply Lemma 30 to the family H defined as follows: α} and the infinite set B α .In this way we get a block sequence ).This ensures that conditions (2) and (4) hold, also, as A α is a block sequence of elements of B α , condition (5) holds.Now, find (and this one exists by Lemma 15).This finishes the construction. Consider ] n has closure of size less than c in F (A) and, as b = c, it has closure of size less than b, so by Corollary 14, it has a convergent subsequence.On the other hand, condition (5) implies that condition (v) of Lemma 27 holds and the other three conditions when f = G are clear, so F (A) is not (n + 1)-sequentially compact.
Parametrized diamonds were defined by Dzamonja, Hrušák and Moore in [41].These principles are weak versions of the well known Jensen's diamond principle ♦ but they have the advantage that are consistent with ¬CH.Moreover, many constructions that follow from ♦ can be carried with an appropriated parametrized diamond.These weak diamond principles can be defined from abstract cardinal invariants.Following [53], we say that triple (A, B, E) is an invariant if A and B are sets of cardinality at most continuum and E is a relation between A and B, such that every element in A is E-related with some element in B and for every b ∈ B there is an a ∈ A such that (a, b) / ∈ E. The evaluation A, B, E of an invariant (A, B, E) is defined as follows: An invariant (A, B, E) is called a Borel invariant [4] if A, B and E are Borel subsets on some Polish space.Given a Borel invariant (A, B, E), its parametrized diamond principle is defined as follows: There exists also a notion of parametrized diamond where the invariant and the function F above do not have the constraint of being Borel.This notion is denoted by Φ(A, B, E), nevertheless, most of the applications of these diamond principles have been shown to follow from its definable counterpart, and unlike them, the definable ones hold in many models in which the associated cardinal invariant is small.
We say that two invariants (A, B, E) and (A ′ , B ′ , E ′ ) are Borel Tukey equivalents (denoted by The general notion of Tukey reductions not restricted to Borel sets (often denoted by ≤ T ) was defined in [53].Since we are only dealing with Borel invariants, we will use ≤ T and ≡ T for the Borel versions.
The importance of Tukey relations between invariants in our context relies in the following result: We will prove that ♦(b) is equivalent to ♦(P, ω ω , ) by showing that (ω ω , ω ω , * ) ≡ T (P, ω ω , ).Proof.Let us begin by noting that (P, ω ω , ) is a Borel invariant.We can identify P as a subset of ω ω if we add {−1} to the range of the functions and assume that f (n) = −1 whenever f is undefined on n.Formally, let W = ω ∪ {−1}.Clearly W ω ≃ ω ω and and then is a Borel relation.
For the other inequality define φ : P → ω ω such that φ(f Clearly ψ is Borel.Before showing that φ es Borel, let us prove that φ and ψ work. Assume φ(f ) * g.Let A = {n ∈ ω : g(n) > φ(f )(n)} which is an infinite set.Given n ∈ A and n defined as above: Thus Â = {n : n ∈ A} is infinite and shows that f ψ(g)(n).
It remains to show that φ is Borel.For this notice that {S n,m : n, m ∈ ω}, where S n,m = {f ∈ ω ω : f (n) = m}, is a subbase for the topology on ω ω .It then suffices to see that φ −1 [S n,m ] is Borel for every n, m ∈ ω.But this is clear since By Proposition 34 the corresponding diamond principles are equivalent.
We are now ready to state the parametrized diamond we will use, in the right representation for our purposes: We call the function g in the previous statement, a ♦(b)-sequence, and we will say that g guesses f at α via F if F (f ↾ α) * g(α).We will often omit mentioning the function F when no confusion arises.
The existence of n-sequentially compact spaces that fail to be (n+1)-sequentially compact under b = c can be adapted to obtain the same counterexamples with size ω 1 under ♦(b).However, even though we can achieve the recursion that guarantees that the space is not (n + 1)-sequentially compact using the guessing property of the ♦(b)-sequence, we also have to ensure that for every sequence f : [ω] n → ω n+1 , there is a subset X ∈ [ω] ω such that the closure of f ′′ [X] n has size less that b (hence it has countable closure since ♦(b) implies b = ω 1 ).We can not achieve this by using the guessing property since we will need every of these functions being guessed cofinally.To avoid this trammel, we will make use of the fact that d is the cofinality of the ideal FIN n .Moreover, under the assumption b = d we can even get a cofinal family in FIN n with stronger properties.
We will use the notation preceding Lemma 30 for the following result.In particular, we will use again Hechler trees and the same enumeration {s n : n ∈ ω} for ω n+1 .Lemma 36.If b = d, there is a family {T (α) : α < d} of Hechler trees in ω ≤n+1 such that In particular there is a base for FIN n+1 of size d.
Proof.Let {d α : α < d} be a scale.As mentioned in the introduction, this family exists since b = d.
For (2) fix X ∈ FIN n+1 and let T a Hechler tree disjoint from X.There is For the last observation let H(α) = ω n+1 \ T (α) and for every For the last part of the previous lemma we do not need b = d, in general we have the following: Proof.By the proof of part 2 in the previous Lemma, we have a cofinal family in FIN n+1 of size d.Thus cof(FIN n+1 ) ≤ d.
Recall that we identify ω n+1 with the set of increasing sequences of length n + 1 and given . For every n ≥ 1, there exists an n-sequentially compact non (n + 1)-sequentially compact space of size ω 1 .
Proof.Fix n ∈ ω and let {T (α) : ω ≤ α < ω 1 } as in Lemma 36.We will recursively construct an almost disjoint family {A α : α < ω 1 } ⊆ ω n+1 .For every infinite α < ω 1 let α = {α m : m ∈ ω}.We will define a Borel function F : 2 <ω1 → P. By a suitable coding we can assume that the domain of F is the set of pairs (B, {A β : β < α}) where It is not hard to see that F is a Borel function.Let g : ω 1 → ω ω be a ♦(b)sequence with the incarnation of b mentioned earlier.Now for every n ∈ ω let A n = C n and for every infinite α < ω 1 define We can assume that every A α is infinite by increasing g(α) if necessary.It is clear from the definition that A = {A α : α < ω 1 } is an almost disjoint family.We will now see that Lemma 15 and by Lemma 36 part 2, there exists α < ω 1 such that for every β > α there is for every infinite β, this set is finite and f ′′ [X] n is almost disjoint with all but countable many A β .By Corollary 13, F (A) is n-sequentially compact.
To see that it is not (n + 1)-sequentially compact let G : [ω] n+1 → ω n+1 where G maps each set s to its increasing enumeration.Thus G ′′ [B] n+1 = B ↑ n+1 for every B ∈ [ω] ω .We will omit the subindex n + 1.Let B ∈ [ω] ω , we have to show that B is not a convergent subsequence for G.This is equivalent to show that there exists α < ω 1 such that A α ∩ B ↑ is infinite by Lemma 27.
Given (B, {A α : α < ω 1 }) coded by f ∈ 2 ω1 , we can assume that {B, {A β : β < α}} is coded by f ↾ α by passing to the club of idecomposable ordinals in ω 1 .Since (B, {A α : α < ω 1 }) is coded by some f ∈ 2 ω1 , there exists α < ω 1 infinite such that We use h as a shorthand for F ((B, {A β : β < α})).Hence there are infinitely many i ∈ B such that g(α)(i) > h(i).Let B ′ be this set and thus C(i, h(i)) ⊆ C(i, g(α)(i)) for every i ∈ B ′ .By the definition of h, there exists As the family {C(i, g(α Another property of the parametrized diamonds principles is that they turn the corresponding cardinal invariant small.For instance, ♦(d) ⇒ d = ω 1 .Thus we can state the previous theorem under ♦(d).In Section 8, we will produce the same examples under s = b.Although ♦(d) (or even d = ω 1 ) implies s = b, a feature of the space constructed in this section is that it has size ω 1 which does not follow from Theorem 52.
Question 39.Does ♦(b) imply that for every n ≥ 1 there is an almost disjoint family A such that F (A) is n-sequentially compact but not (n + 1)-sequentially compact?

High dimensional splitting-like cardinal invariants
We say that P = {P n : n ∈ ω} ⊆ [ω] <ω is an interval partition if P is a partition of ω consisting of consecutive intervals.
Definition 40.Let P = {P n : n ∈ ω} be an interval partition and S ⊆ ω.We say that S block-splits P if A family S ⊆ [ω] ω is a block-splitting family if for every interval partition P, there exists S ∈ S such that S block-splits P. We also define bs = min{|S| : S is a block splitting family}.
It is easy to see that every block-splitting family is a splitting family.It follows that s ≤ bs.In fact, we have the following theorem due to Kamburelis and Węglorz.

Theorem 41. [33] bs = max{b, s}
Recall that for any s ∈ ω ≤n , we denote by C s the set of t ∈ ω n such that t ↾ dom(s) = s.If n ≥ 1 let us denote by L n the set of functions f : [ω] n → ω n+1 such that there is no X ∈ [ω] ω so that f ′′ [X] n ⊆ C s for some s ∈ ω <n+1 \ {∅} (note that it is equivalent if we require this for s ∈ ω 1 ).Definition 42.Let n ≥ 1, f ∈ L n and S ∈ [ω] ω .We say that S splits f (or more precisely L n -splits f ) if there are disjoint X, Y ∈ [ω] ω such that: (1) We define µ n as the minimum size of an L n -splitting family, i.e., the minimum size of a family S ⊆ [ω] ω such that every f ∈ L n is L n -split by an element in S.
At the moment, it is not even clear that there are L n -splitting families, but we will show later in this section that they do exist and find an upper bound for µ n .
We can consider ω n as the set of increasing sequences of ω of length n through the computable bijection ϕ that maps (i 0 , . . ., i n−1 ) into Notation 43.For a set X ⊆ ω and n ∈ ω let We start by finding a lower bound for this new invariants.Proposition 44.s ≤ µ 1 .
Proof.Let S be an L 1 -splitting family of size µ 1 .With this, we will define a splitting family of size at most µ 1 .For every There exists S ∈ S that L 1 -splits f X .Hence there are disjoint Y, Z ∈ [ω] ω such that f ′′ Y ⊆ S 2 and f ′′ Z ⊆ (ω \ S) 2 .In particular, for every n ∈ Y , we know that f (n) ∈ S 2 , which implies that 2x n ∈ S. Similarly, 2x m ∈ ω \ S for every m ∈ Z.
Thus, we can define S ′ = S ∩ 2ω and S = S ′ 2 for every S ∈ S. It is now easy to see that { S : S ∈ S} is a splitting family of size at most µ 1 .
Proof.Let S be an L n+1 -splitting family and let f ∈ L n .Define f ′ ∈ L n+1 such that f ′ maps the cone C s into f (s) k where k is the minimum i that can be concatenated to the sequence f (s).Formally, We now give an upper bound for µ n and implicitly show that L n -splitting families exists.In order to achieve this, we need to work with equivalence relations on [ω] n and apply Erdös-Rado theorem.
Proof.Fix n ∈ ω.We will show that every block splitting family is We claim that I j = ∅ for every j ≤ n.Indeed, if I j = ∅ for some j, there exists k ∈ ω such that f (a) j = k for every a ∈ [M ] n .Since f (a) is increasing for every a ∈ [ω] n , there is a finite set F such that f (a) 0 ∈ F for every a [M ] n , thus applying Ramsey's theorem, we find For every j ≤ n, let m j = max(I j ).As none of the I j is empty, the set {f (a) j : a ∈ [M ] n } is infinite.Moreover, ( * ) for every j ≤ n and k ∈ ω, there exists z ∈ M such that f (a) j > k for every a ∈ [M ] n with z ∈ a = {a 0 , . . ., a n−1 } and z = a i for some i ≤ m j .To see this, fix j ≤ n and k ∈ ω.For every k ′ ≤ k there exists a(k It is also worth noting that given j ≤ n, z ∈ M and t = (t 0 , . . ., t mj−1 ) ∈ (M ∩ z) mj , there is r(j, z, t) ∈ ω such that ( * * ) f (t {z} s) j = r(j, z, t) for every s ∈ [M \ (z + 1)] n−mj (where t s is the concatenation of the sequences t and s).
We are going to define an interval partition P = {P l : l ∈ ω} such that any S, that block splits P, will also L n -split f .It is clear that if we achieve this, then µ n ≤ bs.
To define this partition P, we will also define an increasing sequence {x l : l ∈ ω} ⊆ M and an increasing sequence {k l : l ∈ ω}.
Let x 0 = min(M ) and k 0 = 0.For l > 0, by ( * ), we can find x l > max{x i : i < l} in M such that f (a) j > k l−1 for any j ≤ n and any a ∈ [M ] n such that a i = x l for some i ≤ m j .Given j ≤ n and t = {t 0 , . . ., t mj −1 } ⊆ {x i : i < l} let r(j, x l , t) as in ( * * ) and define Define P l = (k l−1 , k l ] for every l ∈ ω and let P = {P l : l ∈ ω}. We proceed to show that P is as required.Let S ∈ [ω] ω that block-splits P and define X = {x l : P l ⊆ S} and Y = {x l : P l ∩ S = ∅}.Let a = {x li : i < n} ⊆ X where x l0 < x l1 < • • • < x ln−1 .We will prove that f (a) ∈ S n+1 .Fix j ≤ n and let i = m j < n.By the choice of l i , it follows from the definition of x li that f (a) j > k li−1 .On the other hand, we have that f (a) j = f (t x li s) j = r(j, x li , t) < k li with t = {x l0 , . . ., x lm j −1 } and s = {x lm j +1 , . . ., x ln−1 }.Thus f (a) j ∈ P li and P li ⊆ S since x li ∈ X.As this is true for every j ≤ n, we have that f (a) ⊆ S n+1 and therefore It follows from the results in this section that s = µ n = b is consistent for every n ≥ 1.We do not know if in ZFC one of these equalities hold.
We ask the following natural questions: Question 48.It is true that µ n = bs for all n ≥ 1?
8. n-sequentially compact spaces from s = b.
In this section we construct special almost disjoint families following the ideas and techniques developed by Shelah in [49] and improved by Mildenberger, Raghavan and Steprāns in [40] and [46].For a survey of this method see [28], [19].We first need to prove that every L n -splitting family is everywhere L n -splitting.
Recall that a set Y ⊂ ω is almost monochromatic (or almost homogeneous) for a partition f : The cardinal par n denotes the smallest cardinal κ such that there is a family of partitions of [ω] n of size κ such that no infinite set is almost monochromatic for all of them simultaneously.
Note that if we consider partitions into some finite number of pieces k, instead of 2 pieces, we obtain the same cardinal.Moreover, for all n ≥ 2 we have that par n = par 2 , in fact: If S ⊆ ω then we will follow the notation S 0 := S and S 1 := ω \ S. We are now ready to prove the main theorem in this section.It is interesting that the combinatorial characterizations of both max{b, s} and min{b, s} are used in the proof of this theorem.
Theorem 52.(s = b) For every n ≥ 1 there exists an almost disjoint family A on ω n+1 of size c such that F (A) is n-sequentially compact but not (n + 1)-sequentially compact.
Proof.Let us fix S = {S α | α ∈ b} that is splitting and L n -splitting and also let us fix the following enumerations: As usual let G : [ω] n+1 → ω n+1 be the increasing enumeration.
We want to construct the following families: with the following properties: (1) ( Let us suppose that we have already constructed the desired objects until step α.We are going to define X α and σ α .
-X α and σ α construction: We are going to construct As S is L n -splitting, then we can find n+1 for i ∈ 2. Now consider the following family of colorings on [Z 0 ∪ Z 1 ] n : for every β < α ∅ let g β : [Z 0 ∪ Z 1 ] n → 3 as follows: Claim 53.For every β < α ∅ there exists j ∈ 2 such that for every i ∈ 2 we have Proof.Consider Y 0 and Y 1 and the coloring g β .If Y i is almost 0-monochromatic for any i ∈ 2 then Y 1−i can not be almost 1-monochromatic, since S β does not split f .In this case take j = 0. Hence 1 − j = 1 and no matter which color Y 1−i takes, we know that there exists β ) n+1 = ∅ for both i and i − 1. Analogously, if there is an i < 2 such that Y i is 1-monochromatic, j = 1 works.In case both sets are 2-monochromatic, any j ∈ 2 satisfies the claim.Now define η ∅ : α ∅ → 2 such that for every β < α ∅ we have that η ∅ (β) = j, where j is given by the claim.For f (0) and f (1) we just take the restriction Now suppose that we have already constructed f s , Y s , α t and η t : α t → 2 for s = t j where j ∈ 2 and t ∈ 2 <ω .We are going to construct Y s 0 , Y s 1 , α s and η s . As , there exists a minimum α s ∈ b such that S αs splits f s .Note that α s > α t since no β ≤ α t splits f s .Thus there exist Z for every i ∈ 2. As before, consider the following family of colorings of for some m β ∈ ω and every i ∈ 2.
On the other hand, we know that Y s 0 , Y s 1 ⊆ Y s and our induction hypothesis = ∅ for some m ∈ ω, so this last one condition is also true for Y s 0 , Y s 1 .Finally, let This shows that condition (f) holds and all other conditions are clear from the the construction.
We are going to construct a collection .The latter, the fact that S αs does not split B s i and the fact that B s ⊆ B t for t ⊆ s, imply that η s η s i for every i ∈ 2. Now, for every g ∈ 2 ω let η g = n∈ω η g↾n .As before η g ∈ 2 <b and if f = g, η f and η g are incompatible nodes of 2 <b .As α < c, we can find g ∈ 2 ω such that there is no β < α such that σ β or τ β extends η g , then let Note that if β < α and σ α and τ β are incompatibles, then there exists m ∈ ω Consider the following families of elements of FIN n+1 : Note that |H| < b, so we can apply Lemma 30 to the family H and the infinite set Z to obtain A α a block sequence of elements of Z such that its intersection with every element of H is finite.Given β ∈ dom(τ α ), there is s ∈ 2 <ω such that β ∈ dom(η s ) and s ⊆ g.Thus Let us verify that this construction satisfies conditions (1) to (9).Conditions (1), ( 2), ( 4), ( 5) are true by definition.Condition (6) holds since A α is a block sequence of elements of Z ⊆ B α .Condition (8) follows from the choice of σ α .We have already checked condition (3) at the end of the construction of A α .It remains to show that (7) and ( 9) hold.Condition (7): Let β and α in c such that ω ≤ β < α.If σ β ⊆ τ α , then for the construction of A α we know that f On the other hand if σ β and τ α are incompatibles then exists ξ ∈ dom(τ α ) ∩ dom(σ β ) such that τ α (ξ) = i and σ β (ξ) = 1 − i.By (3) and ( 8) applied to ξ we get the result.
Condition (9): It is clear that if i = j are finite ordinals then Let us see that F (A) is the desired space.Conditions (3) and ( 6) imply that for every infinite α ∈ c, the elements of the almost disjoint family in the closure of f ′′ α [X α ] n , are at most those A β such that τ β ⊆ σ α .In particular, the closure of f ′′ α [X α ] n has size less than b.By Corollary 14 we get that F (A) is n-sequentially compact.On the other hand, condition (7) imply that condition (v) of Lemma 27 holds and the other three conditions when f = G are clear, so F (A) is not (n + 1)-sequentially compact.
Most modifications of the technique developed by Shelah used previously in the literature use assumptions of the form ι ≤ θ where ι is a kind of splitting cardinal invariant and θ is another cardinal invariant imposed by the particularities of the problem.We do not know if the previous result can be weakened to s ≤ b.
Question 54.Does it follow from s ≤ b that there are n-sequentially compact spaces that fail to be (n + 1)-sequentially compact?
Or even better Question 55.Is there in ZFC, a space that is n-sequentially compact but no (n+1)sequentially compact for every n ≥ 2?
9. On the character of n-sequentially compact spaces One of the key facts in the constructions of our examples has been that any sequentially compact space of character less than b is n-sequentially compact for every n ≥ 1.It is then natural to ask what is the smallest possible character of an n-sequentially compact space that is not (n + 1)-sequentially compact: Definition 56.If n ≥ 2 let λ n be the minimum character of a space X such that X is (n − 1)-sequentially compact but not n-sequentially compact.
We do not know if these cardinals are well defined since we do not know if the are n-sequentially compact spaces that are not (n + 1)-sequentially compact in ZFC (for n > 1).To avoid unnecessary complications, let us agree that λ n = c when it is not well defined by the previous definition.Theorem 12 gives us b as a lower bound for λ n for every n ≥ 2. Now, we will see that, using the following space, introduced by Burke and van Douwen in [7], we can actually compute λ 2 .
Let {f α | α ∈ b} consisting of strictly increasing functions, which is well ordered by ≤ * and for each α b} and let X = (ω × ω) ∪ B be a topological space, where the elements of ω × ω are isolated and a basic neighborhood for B β ∈ B has the form It is easy to check that X is Hausdorff, separable and since the basic neighborhoods of every point are compact, then X is also locally compact (see Example 7.3 of [52]).On the other hand, X is not sequentially compact since for all n ∈ ω, the sequence ((n, m)) m∈ω does not converge in X.So let X = X ∪ {∞} be the one-point compactification of X.Then the basic neighborhoods of ∞ have the form: where C ⊆ X is compact.
Theorem 57.The space X is a compact and sequentially compact space of character b that is not 2-sequentially compact.
Proof.Let us see that X is sequentially compact.Let f : ω → X.If f takes the value ∞ infinitely many times, then we are done.Also if f takes values in B infinitely many times, then we are done since b has uncountable cofinality (and then it is sequentially compact).So let us suppose that f : every basic neighborhood of ∞ contains all but finite elements of A n for every n ∈ ω.
We will now see that X is not 2-sequentially compact, so consider G : [ω] 2 → X given by G({n, m}) = (n, m) where n < m.We will see that for all We can still ask for the minimal size of an almost disjoint family A such that F (A) is not 2-sequentially compact, i.e., the minimal character of a space of the form F (A) that is not 2-sequentially compact.In general we can ask this for every n ≥ 2: Definition 59.If n ≥ 2 let a sc n be the minimum size of an almost disjoint family such that F (A) is (n − 1)-sequentially compact but not n-sequentially compact.
It follows from the results in [37] that b ≤ a sc 2 ≤ a.Also, it should be clear that λ n ≤ a sc n for all n ≥ 2 whenever they are well defined.Definition 60. [49] Let a * be the minimal size of an ad family A, such that there are (disjoint) C 0 , C 1 , C 2 , . . .∈ A ⊥ so that for every D ∈ [ω] ω , if |C n ∩ D| = ω for infinitely many n ∈ ω, then there is A ∈ A for which |A ∩ D| = ω.
The following notions were introduced by Arhangel'skii [1]: Definition 61.Let X be a topological space and x ∈ X.Then x is an α i -point (i = 1, 2, 3) if for every family {S n | n ∈ ω} of sequences from X converging to x, there exists a sequence S that converges to x such that: • (α 1 ) S n \ S is finite for all n ∈ ω, • (α 2 ) S ∩ S n = ∅ for all n ∈ ω.
• (α 3 ) |S ∩ S n | = ω for infinitely many n ∈ ω.A space X is an α i -space if every point x ∈ X is an α i -point.
Arhangel'skii introduced the α i properties for i ∈ {1, 2, 3, 4, 5} in [1].It is easy to see that all α 1 -points are α 2 -points and that all α 2 -points are α 3 -points.We refer the reader to [1] for the rest of the definitions and a deep study on their relationship.
Proposition 62. a * is the minimum size of an ad family such that F (A) is not α 3 .
Proof.Let κ < a * and be A ad family of cardinality κ.As we want to prove that F (A) is α 3 , it is enough to prove that ∞ is an α 3 point since any other point has countable character.Let {S n | n ∈ ω} be a collection of convergent sequences to ∞.We can assume that S n only takes values in ω for every n ∈ ω, since any sequence in A converges to ∞.As S n is convergent to ∞, S n ∈ A ⊥ for every n ∈ ω.Since |A| < a * , there exists B ∈ [ω] ω such that S n ∩ B is infinite for infinitely many S n and A ∩ B is finite for every A ∈ A. Then B is the required convergent sequence to ∞.
Conversely, let A be an ad family such that F (A) is α 3 .We want to prove that A is not a witness of a * , so let C 0 , C 1 , C 2 , . . .∈ A ⊥ .Note that every C n is a convergent sequence to ∞ and as ∞ is an α 3 -point, there exists a sequence B ⊆ Ψ(A) that converges to ∞ and such that |C n ∩ B| = ω for infinitely many n ∈ ω.We can assume that B ⊆ ω since otherwise B ′ = B ∩ ( i∈ω )C i is also a witness for the α 3 -property for the family {C i | i ∈ ω}.As B converges to ∞, we also have that B ∩ A = * ∅ for all A ∈ A, so A is not a witness for a * as we wanted to show.
Following the idea of the previous proof, we can actually show that 2-sequentially compact ad families are α 3 .Given a non α 3 ad family A and a family {S n : n ∈ ω} ⊆ A ⊥ that witnesses this, we can define f : [ω] 2 → ω by f ({n, m}) = S n (m), where S n (m) is the m th element of S n .Thus A and A 0 = {S n | n ∈ ω} satisfy the hypothesis of Lemma 27 for n = 2 and F (A) is not 2-sequentially compact.
In general, it is possible to prove that 2-sequentially compact spaces are α 3 .This result was independently proved by Szeptycki, Almontashery and Memarpanahi.Moreover, they also proved that 2-sequentially compact spaces do not need to be α 2 and that α 1 spaces need not to be 2-sequentially compact.
and the last is finite.We conclude that Y converges to x.
It is also worth to mention that the previous proof follows the same idea as the proof that the spaces with the Ramsey property considered by Knaust [35] are α 3 .Recall that a space X has the Ramsey property if whenever lim i→∞ lim j→∞ x(i, j) = x, there exists B ∈ [ω] w such that f ↾ [B] 2 converges to x, where f ({i, j}) = x(i, j).
Corollary 64.If A is an ad family such that F (A) is 2-sequentially compact, then F (A) is α 3 .
In [21] it is proved that b ≤ a * ≤ a and Corollary 64 implies that a sc 2 ≤ a * , so we have that b = λ 2 ≤ a sc 2 ≤ a * ≤ a.Of course, the inequality λ n ≤ a sc n is clear from the definition.This is pretty much what we know so far about these cardinals.
Question 65.Are there any other relations between the cardinals λ n and a sc n with b and a?
Question 66.Is it consistent that a sc 2 < a * ?Question 67.Are all the λ n and a sc m consistently different?In particular, is

Theorem 16 . [ 12 ]
For every equivalence relation E on [ω] k there are M ∈ [ω] ω and I ⊆ k such that aEb if and only if a ↾ I = b ↾ I.
this will suffice.Let m = max F .Since A ∩ A i is finite for every i ≤ m and |f ′′ [B] n ∩ A| = ω, there exists a ∈ [B \ F ] n such that f (a) ∈ A and we are done.
5. n-sequentially compact spaces from b = c.
almost monochromatic for g β for every β < α ∅ .It is worth to point out that we are defining η ∅ in terms of Y 0 = Y ∅ 0 and Y 1 = Y ∅ 1 .

2
does not converge in X.So let us fix B ∈ [ω] ω and B β ∈ B. Note that for every basic neighborhood N of B β we have that N ∩ A n is finite for every n ∈ ω, but G ′′ [B \ F ] 2 has infinite intersection with every A n such that n ∈ B \ F .This shows that there is no F ∈ [ω] <ω such that G ′′ [B \ F ] 2 ⊆ N .On the other hand, as G is injective, G ↾ [B] 2 does not converge to any isolated point.So the only thing that remains to be seen is that G ↾ [B] 2 does not converge to ∞.Let (m k ) k∈ω be the increasing enumeration of B and letA = {(m k , m k+1 ) | k ∈ ω}.Note that: (1) A ∈ A. (2) For all F ∈ [ω] <ω we have that A ⊆ * G ′′ [B \ F ] 2 .Now let B β ∈ B such that |B β ∩ A| = ωand β is the minimum with this property.Consider the following basic neighborhood of ∞:N = {∞} ∪ (X \ N (B −1 , B β , ∅)) = {∞} ∪ (B \ (B −1 , B β ]) ∪ ((ω × ω) \ B β ) .Remember that B −1 = ∅.If there exists F ∈ [ω] <ω such that G ′′ [B \ F ] 2 ⊆ N , then in particular we have that G ′′ [B \ F ] 2 ∩ B β = ∅, and as A ⊆ * G ′′ [B \ F ] 2 , then A ∩ B βis finite, which is a contradiction.This implies that there is no such F and then G ↾ [B] 2 does not converge, which proves that X is not 2-sequentially compact.With this last result and Theorem 12, we get the following nice topological characterization of b:Corollary 58.b = λ 2 , i.e.: b = min{χ(X) | X is sequentially compact but not 2-sequentially compact}.

Theorem 63 .
If X is 2-sequentially compact, then X is α 3 .Proof.Let x ∈ X and for every n ∈ ω let Y n = {x(n, m) : m ∈ ω} be a sequence converging to x. Define f : [ω] 2 → X by f (n, m) = x(n, m), where n < m.Since X is 2-sequentially compact, there exists B ∈ [ω] ω such that f ↾ [B] 2 converges.Let Y := f ′′ [B] 2 ∈ [X] ω .It is clear that Y ∩ f ′′ n ω is infinite for every n ∈ B, so we are done if we prove that Y is convergent to x.Let U be an open set such that x ∈ U .Since f ↾ [B] 2 converges to x, we know that there exists m ∈ ω such that f ′′ [B \ m] 2 ⊆ U .It follows that Y \ U ⊆ n∈B∩m Y n .But n∈B∩m Y n is a convergent sequence to x, so n∈B∩m Y n \ U is finite, then so there exists Y s i ∈ [Z i ] ω such that Y s i is almost-homogeneous for every β ∈ [α t , α s ).At this point we can define η ′ s : [α t , α s ) → 2 with the proper adaptation of Claim 53.Now let η s = η t ∪ η ′ s .From the definition of η ′ s , we know that for every β ∈ [α t , α s ), then B s ⊆ * S Suppose that we have already constructed η t and α t for every t s and B s .Letα s ∈ b be the minimum such that |B s ∩ S 0 αs | = ω = |B s ∩ S 1αs |, this is possible since S is splitting.Notice also that α t < α s whenever t s by (vi) and (v).Let η s : α s → 2 such that for all β < α s , |B s ∩ S Finally, for every i ∈ 2 let B s i = B s ∩ S i αs .Note that for every β ∈ α s and for i ∈ {0, 1}, we have that B s i is almost contained in S