TWO-CARDINAL DERIVED TOPOLOGIES, INDESCRIBABILITY AND RAMSEYNESS

. We introduce a natural two-cardinal version of Bagaria’s sequence of derived topologies on ordinals. We prove that for our sequence of two-cardinal derived topologies, limit points of sets can be characterized in terms of a new iterated form of pairwise simultaneous reﬂection of certain kinds of stationary sets, the ﬁrst few instances of which are often equivalent to notions related to strong stationarity, which has been studied previously in the context of strongly normal ideals [10]. The non-discreteness of these two-cardinal derived topologies can be obtained from certain two-cardinal indescribability hypotheses, which follow from local instances of supercompactness. Additionally, we answer several questions posed by the ﬁrst author, Peter Holy and Philip White on the relationship between Ramseyness and indescribability in both the cardinal context and in the two-cardinal context.


Introduction
The derived set of a subset A of a topological space (X, τ ) is the collection d(A) of all limit points of A in the space.We refer to the function d as the Cantor derivative of the space (X, τ ).Recently, Bagaria showed [2] that the derived topologies on ordinals, whose definition we review now, are closely related to certain widely studied stationary reflection properties and large cardinal notions.Suppose δ is an ordinal and τ 0 is the order topology on δ.That is, τ 0 is the topology on δ generated by B 0 = {{0}} ∪ {(α, β) | α < β < δ}.For a set A ⊆ δ, it easily follows that the collection d 0 (A) of all limit points of A in the space (δ, τ 0 ), is equal to {α < δ | A is unbounded in α}.Beginning with the interval topology on δ and declaring more and more derived sets to be open, Bagaria [2] introduced the sequence of derived topologies τ ξ | ξ < δ on δ.For example, τ 1 is the topology on δ generated by B 1 = B 0 ∪ {d 0 (A) | A ⊆ δ}, and τ 2 is the topology on δ generated by B 2 = B 1 ∪ {d 1 (A) | A ⊆ δ} where d 1 is the Cantor derivative of the space (δ, τ 1 ).Bagaria showed that limit points of sets in the spaces (δ, τ ξ ), for ξ ∈ {1, 2}, can be characterized as follows.For A ⊆ δ and α < δ: α is a limit point of A in (δ, τ 1 ) if and only if A is stationary in α, and α is a limit point of A in (δ, τ 2 ) if and only if whenever S and T are stationary subsets of α there is a β ∈ A such that S ∩ β and T ∩ β are stationary subsets of β.Furthermore, Bagaria proved that limit points of sets in the spaces (δ, τ ξ ) for ξ > 2 can be characterized in terms of an iterated form of pairwise simultaneous stationary reflection called ξ-s-stationarity.
In this article we address the following natural question: is there some analogue of the sequence of derived topologies on an ordinal in the two-cardinal setting?Specifically, suppose κ is a cardinal and X is a set of ordinals with κ ⊆ X.Is there a topology τ ξ on P κ X such that, for all A ⊆ P κ X, the limit points of A in the space (P κ X, τ ξ ) are precisely the points x ∈ P κ X such that the set A satisfies: • some unboundedness condition at x? • some stationarity condition at x? • some pairwise simultaneous stationary reflection-like condition at x? Recall that for x, y ∈ P κ X we say that x is a strong subset of y and write x ≺ y if x ⊆ y and |x| < |y ∩κ|.Let us note that the ordering ≺, and its variants, are used in the context of supercompact Prikry forcings [19].In Section 3.1, we show that the ordering ≺ induces a natural topology τ 0 on P κ X analogous to the order topology on an ordinal δ.Furthermore, beginning with τ 0 and following the constructions of [2], in Section 3.2 we define a sequence of derived topologies τ ξ | ξ < κ on P κ X.Let us note that after submitting the current article, the authors learned that Catalina Torres, working under the supervision of Joan Bagaria, simultaneously and independently defined a sequence of two-cardinal derived topologies and obtained results similar to those in Sections 3.2 -3.6 involving the relationship between various two-cardinal notions of ξ-s-stationarity and two-cardinal derived topologies.
We show (see Propositions 3.10 and 3.14) that in the space (P κ X, τ 1 ), for x ∈ P κ X with x ∩ κ an inaccessible cardinal, x is a limit point of a set A ⊆ P κ X if and only if A is strongly stationary in P x∩κ x (see Section 2 for the definition of strongly stationary set).Let us note that although the notion of strong stationarity is distinct from the widely popular notion of two-cardinal stationarity introduced by Jech [21] (see [12,Lemma 2.2]), it has previously been studied by several authors [10,12,23,24,29].The analogy with the case of derived topologies on ordinals continues: in the space (P κ X, τ 2 ), when x ∈ P κ X is such that x ∩ κ < κ and P x∩κ x satisfies a two-cardinal version of Π 1  1 -indescribability, x is a limit point of a set A ⊆ P κ X if and only if for every pair S, T of strongly stationary subsets of P κ∩x x there is a y ≺ x in A with y ∩ κ < x ∩ κ such that S and T are both strongly stationary in P y∩κ y (see Proposition 3.30).Additionally, using a different method, we show (see Corollary 3.36) that if κ is weakly inaccessible and X is a set of ordinals with κ ⊆ X, then there is a topology on P κ X such that for A ⊆ P κ X, x ∈ P κ X is a limit point of A if and only if κ x is weakly inaccessible and A is stationary in P κx x in the sense of Jech [21].
In order to prove the characterizations of limit points of sets in the spaces (P κ X, τ ξ ) (Theorem 3.16(1)), we introduce new iterated forms of two-cardinal stationarity and two-cardinal pairwise simultaneous stationary reflection, which we refer to as ξ-strong stationarity and ξ-s-strong stationarity (see Definition 3.7).Let us note that the notions of ξ-strong stationarity and ξ-s-strong stationarity introduced here are natural generalizations of notions previously studied in the cardinal context by Bagaria, Magidor and Sakai [4], Bagaria [2] and by Brickhill and Welch [8], as well as those previously studied in the two-cardinal context by Sakai [26], by Torres [28], as well as by Benhamou and the third author [7].
We establish some basic properties of the ideals associated to ξ-strong stationarity and ξ-s-strong stationarity and introduce notions of ξ-weak club and ξ-s-weak club which provide natural filter bases for the corresponding ideals (see Corollary 3.18).The consistency of the non-discreteness of the derived topologies τ ξ on P κ X is obtained using various two-cardinal indescribability hypotheses, all of which follow from appropriate local instances of supercompactness (see Section 3.5).We also show that by restricting our attention to a certain natural club subset of P κ X, some questions about the resulting spaces, such as questions regarding when particular subbases are in fact bases, become more tractable (see Section 3.6).
Additionally, in Section 4, we answer several questions asked by the first author and Peter Holy [15] and the first author and Philip White [16] concerning the relationship between Ramseyness and indescribability.For example, answering [15,Question 10.9] in the affirmative, we show that the existence of a 2-Ramsey cardinal is strictly stronger in consistency strength than the existence of a 1-Π 1  1 -Ramsey cardinal.In other words, the existence of an uncountable cardinal κ such that for every regressive function f : [κ] <ω → κ there is a set H ⊆ κ which is positive for the Ramsey ideal homogeneous for f , is strictly stronger in consistency strength than the existence of a cardinal κ such that for every regressive function f : [κ] <ω → κ there is a set H ⊆ κ that is positive for the Π 1  1 -indescribability ideal and homogeneous for f .

Strong stationarity and weak clubs
Suppose κ is a cardinal and X is a set of ordinals with κ ⊆ X.Given x ∈ P κ X, we denote |x ∩ κ| by κ x .We define an ordering ≺ on P κ X by letting x ≺ y if and only if x ⊆ y and |x| < κ y .An ideal I on P κ X is strongly normal if whenever S ∈ I + and f : S → P κ X is such that f (x) ≺ x for all x ∈ S, then there is some T ∈ P (S)∩I + such that f ↾ T is constant.It is easy to see that an ideal I is strongly normal if and only if the dual filter I * is closed under ≺-diagonal intersections in the following sense: whenever A x ∈ I * for all x ∈ P κ X, the ≺-diagonal intersection is in I * .Carr, Levinski and Pelletier [10] showed that there is a strongly normal ideal on P κ X if and only if κ is a Mahlo cardinal or κ = µ + for some cardinal µ with µ <µ = µ.Furthermore, they proved that when a strongly normal ideal exists on P κ X, the minimal such ideal is that consisting of the non-strongly stationary subsets of P κ X, which are defined as follows.Given a function f : Thus, when κ is Mahlo or κ = µ + where µ <µ = µ, the ideal NSS κ,X is the minimal strongly normal ideal on P κ X.
When κ is Mahlo, we can identify a filter base for the filter dual to NSS κ,X consisting of sets which are, in a sense, cofinal in P κ X and satisfy a certain natural closure property.We say that a set X} generate the same filter on P κ X, namely, the filter NSS * κ,X dual to the ideal NSS κ,X .Proof.By definition, the filter on P κ X generated by C 0 is NSS * κ,X .Let us show that the filter generated by C 1 equals that generated by For the other direction, we fix a function g : We define an increasing chain x η | η < κ in P κ X as follows.Let x 0 = x.Given x η we choose x η+1 ∈ P κ X with κ xη+1 = x η+1 ∩ κ and f [P κx η x η ] ≺ x η+1 .When η < κ is a limit ordinal we let x η = α<η x α .Then κ xη | η < κ is a strictly increasing sequence in κ and the set which implies g(a) ∈ P κx η x η and hence x η ∈ C g .Since x ≺ x η , it follows that C g is ≺-cofinal.Now we verify that C g is ≺-closed in P κ X. Suppose C g ∩ P κx x is ≺-cofinal in P κx x.We must show that x ∈ C g .Suppose y ∈ P κx x.Then there is some z ∈ C g with y ≺ z ≺ x.Thus g(y) ≺ z ≺ x and hence x ∈ C g .Now let us verify that the filter generated by C 0 equals that generated by C 1 .For any function f : P κ X → P κ X we have C f ⊆ B f , so the filter generated by C 0 is contained in the filter generated by C 1 .Let us fix a function g : P κ X → P κ X.We must show that there is a function h : P κ X → P κ X such that B h ⊆ C g .Define h : P κ X → P κ X by letting h(x) be some member of C g with g(x) ≺ h(x), for all x ∈ P κ X. Suppose x ∈ B h .To show x ∈ C g , suppose y ≺ x.Then it follows that g(y) ≺ h(y) ⊆ x, which implies g(y) ≺ x and thus x ∈ C g .Therefore B h ⊆ C g and hence the filter generated by C 0 equals the filter generated by C 1 .
We end this section by discussing the more common variants of "club" and "stationary" subsets of P κ X, introduced by Jech in [21].Recall that, for a regular cardinal κ and a set X ⊇ κ, a set C ⊆ P κ X is said to be club in P κ X if it is ⊆-cofinal in P κ X and, whenever D ⊆ C is a ⊆-linearly ordered set of cardinality less than κ, we have D ∈ C.This latter requirement is equivalent to the following formal strengthening: whenever D ⊆ C is ⊆-directed and |D| < κ, we have D ∈ C. We then say that a set S ⊆ P κ X is stationary if, for every club C in P κ X, we have S ∩ C = ∅.The following basic observation justifies the use of the name "weak club" for the notion thusly designated above.Proposition 2.2.If κ is weakly inaccessible, X ⊇ κ is a set of ordinals, and C is a club in P κ X, then C is a weak club in P κ X.
Proof.Suppose that C is a club in P κ X.Since κ is a limit cardinal, the fact that C is ⊆-cofinal implies that it is also ≺-cofinal.To verify closure, fix

Two-cardinal derived topologies and ξ-strong stationarity
Fix for this section an arbitrary regular uncountable cardinal κ and a set of ordinals X ⊇ κ.We will investigate a sequence of derived topologies τ ξ | ξ < κ on P κ X, simultaneously isolating a hierarchy of stationary reflection principles that characterize the existence of limit points with respect to these topologies.We emphasize that all definitions and arguments in this section are in the context of the ambient space of P κ X.We begin by describing τ 0 , a generalization of the order topology.

3.1.
A generalization of the order topology to P κ X.Given x, y ∈ P κ X with x ≺ y, let Let τ 0 be the topology on P κ X generated by It is easy to see that B 0 is a base for τ Proof.Fix A ⊆ P κ X and x ∈ d 0 (A), and suppose that y ∈ P κx x.Since (y, x] is an open neighborhood of x, we can choose a z ∈ (y, x] ∩ A with z = x.This implies z ∈ (y, x) ∩ A, and hence A is ≺-cofinal in P κx x.Conversely, suppose A is ≺-cofinal in P κx x and let (a, b] be a basic open neighborhood of x.Then a ∈ P κx x and we may choose some y ∈ A with a ≺ y ∈ P κx x.Hence y ∈ (a, b] ∩ A \ {x}.
Corollary 3.2.A point x ∈ P κ X is not isolated in τ 0 if and only if κ x = |x ∩ κ| is a limit cardinal.
The following proposition connects the order topology τ 0 on P κ X to the notion of weak club discussed in Section 2, in the case where κ is a weakly Mahlo cardinal.
Let us show that d 0 (A) is ≺-cofinal in P κ X. Fix x ∈ P κ X.We define an increasing chain x η | η < κ in P κ X as follows.Let x 0 = x.Given x η choose x η+1 ∈ A with x η ≺ x η+1 .If η < κ is a limit let x η = ζ<η x ζ .Then κ xη | η < κ is a strictly increasing sequence in κ and the set Recall that an ordinal δ has uncountable cofinality if and only if for every A ⊆ δ which is unbounded in δ, there is an α < δ such that A is unbounded in α.The following proposition is the analogous result for the notion of ≺-cofinality in P κ X when κ is weakly inaccessible.Proposition 3.4.If κ is weakly inaccessible, then the following are equivalent.
(2) For all A ⊆ P κ X if A is ≺-cofinal in P κ X then there is an x ∈ P κ X such that A is ≺-cofinal in P κx x.
Proof.The fact that (1) implies (2) follows from Proposition 3.3.Let us show that (2) implies (1).We assume (2) holds, and that κ is weakly inaccessible but not weakly Mahlo.Let C ⊆ κ be a club consisting of singular cardinals, and let Let a ⊆ κ y be cofinal in κ y with |a| = cf(κ y ) < κ y .Since y ∩ κ is an ordinal we have a ⊆ κ y = |y ∩ κ| ⊆ y ∩ κ and thus a ∈ P κy y.However, there is no x ∈ D ∩ P κy y with a ≺ x because for such an x, κ ∩ x ∈ C would be an ordinal containing the set a which is cofinal in κ y , and hence κ x ≥ κ y .
We note that the assumption that κ is weakly inaccessible is necessary in Proposition 3.4, but only for the somewhat trivial reason that, if κ is a successor cardinal, then there are no ≺-cofinal subsets of P κ X.

Definitions of derived topologies and iterated stationarity in
With the topology τ 0 on P κ X, the base B 0 for τ 0 and the Cantor derivative d 0 in hand, we can now define the derived topologies on P κ X as follows.Given τ ξ , B ξ and d ξ , we let B ξ+1 = B ξ ∪ {d ξ (A) | A ⊆ P κ X}, we let τ ξ+1 be the topology generated by B ξ+1 and we let d ξ+1 be defined by for A ⊆ P κ X.When ξ is a limit ordinal we let τ ξ be the topology generated by B ξ := ζ<ξ B ζ and we let d ξ be the Cantor derivative of the space (P κ X, τ ξ ).
Since B 0 is a base for τ 0 , it easily follows that the sets of the form where I ∈ B 0 , n < ω, ξ i < ξ and A i ⊆ P κ X for i < n, form a base for τ ξ .We return to the question of whether or not B ξ forms a base for τ ξ in Theorem 3.22 below, as well as in Subsection 3.6.
Let us note here that the next two lemmas can easily be established using arguments similar to those for [2, Proposition 2.1 and Corollary 2.2].Lemma 3.5.For all ζ < ξ and all A 0 , . . ., A n ⊆ P κ X, Lemma 3.6.For every ordinal ξ, the sets of the form where I ∈ B 0 , n < ω, ζ < ξ and A i ⊆ P κ X for i < n, form a base for τ ξ .
In the next few sections, we will characterize the non-isolated points of the spaces (P κ X, τ ξ ) in terms of the following two-cardinal notions of ξ-s-strong stationarity.Definition 3.7.
(1) For A ⊆ P κ X and x ∈ P κ X, we say that A is 0-strongly stationary in P κx x if and only if A is ≺-cofinal in P κx x.For an ordinal ξ > 0, we say that A is ξ-strongly stationary in P κx x if and only if κ x is a limit cardinal 1 and, whenever ζ < ξ and S ⊆ P κx x is ζ-strongly stationary in P κx x, there is some y ∈ A ∩ P κx x such that S is ζ-strongly stationary in P κy y.
(2) A set C ⊆ P κ X is called 0-weak club in P κx x if and only if it is ≺-cofinal and ≺-closed in P κx x.For an ordinal ξ > 0, we say that C is ξ-weak club in P κx x if and only if it is ξ-strongly stationary in P κx x and it is ξ-strongly stationary closed in P κx x, meaning that whenever y ≺ x and C is ξ-strongly stationary in P κy y we have y ∈ C. (3) We say that A is 0-s-strongly stationary in P κx x if and only if A is ≺-cofinal in P κx x.For an ordinal ξ > 0, we say that A is ξ-s-strongly stationary in P κx x if and only if κ x is a limit ordinal and, whenever ζ < ξ, κ x and S, T ⊆ P κ X are ζ-s-strongly stationary in P κx x there is some y ∈ A ∩ P κx x such that S and T are both ζ-s-strongly stationary in P κy y. (4) A set C ⊆ P κ X is called 0-s-weak club in P κx x if and only if it is 0-s-strongly stationary in P κx x and whenever y ≺ x and C is 0-s-strongly stationary in P κy y we have y ∈ C. For an ordinal ξ > 0, we say that C is ξ-s-weak club in P κx x if and only if it is ξ-s-strongly stationary in P κx x and it is ξ-s-closed 1 The requirement that κx be a limit cardinal in order for A to be ξ-strongly stationary in Pκ x x is necessary because otherwise, when κx is a successor ordinal there are no 0-strongly stationary subsets of Pκ x x and hence every subset of Pκ x x would be 1-strongly stationary.
in P κx x, meaning that whenever y < x and C is ξ-s-strongly stationary in P κy y we have y ∈ C.
In what follows, given x ∈ P κ X and ξ < κ, we will simply say that, e.g., P κx x is ξ-s-strongly stationary to mean that it is ξ-s-strongly stationary in P κx x.Let us first note the following simple proposition, which justifies the restriction of our attention to values of ξ less than κ.By the results of Subsection 3.5, the proposition is sharp, at least assuming the consistency of certain large cardinals.Proposition 3.8.For all x ∈ P κ X, P κx x is not (κ x + 1)-strongly stationary.
Proof.Suppose otherwise, and let x ∈ P κ X be a counterexample such that κ x is minimal among all counterexamples.Since P κx x is (κ x + 1)-strongly stationary, it is a fortiori κ x -strongly stationary.Therefore, by the definition of (κ x + 1)-strongly stationary, we can find y ∈ P κx x such that P κx x is κ x -strongly stationary in P κy y.Since κ x > κ y , this implies that P κy y is (κ y + 1)-strongly stationary, contradicting the minimality of κ x .
Considering the previous proposition, it is natural to wonder whether the definitions of ξ-strong stationarity and ξ-s-strong stationarity can be modified using canonical functions to allow for settings in which some x ∈ P κ X can be ξ-strongly stationary for κ x < ξ < |x| + ; this was done in the cardinal setting by the first author in [11].See the discussion before Question 5.7 and Question 5.8 for more information.
Definition 3.7 leads naturally to the definition of the following ideals, which can be strongly normal under a certain large cardinal hypothesis by Proposition 3.31.Definition 3.9.Suppose that x ∈ P κ X.We define NS ξ κx,x = {A ⊆ P κ X | A is not ξ-strongly stationary in P κx x} and NS ξ κx,x = {A ⊆ P κ X | A is not ξ-s-strongly stationary in P κx x}.Let us show that for the x's in P κ X that we will care most about, namely those for which κ x is regular, 1-strong stationarity and 1-s-strong stationarity are equivalent in P κx x; moreover, if κ x is inaccessible, then these notions are equivalent to strong stationarity in P κx x plus the Mahloness of κ x .Proposition 3.10.Suppose A ⊆ P κ X and x ∈ P κ X with κ x regular.Then the following are equivalent, and both imply that κ x is weakly Mahlo.
(2) A is 1-s-strongly stationary in P κx x.If, moreover, κ x is strongly inaccessible, then these two statements are also equivalent to the following: (3) κ x is Mahlo and A is strongly stationary in P κx x.
Proof.Note that, if A is 1-strongly stationary in P κx x, then κ x is a limit cardinal and hence weakly inaccessible.We can thus assume that this is the case.( 2) =⇒ (1) is trivial.Let us now assume that A is 1-strongly stationary in P κx x.By Proposition 3.4, it follows that κ x is weakly Mahlo.To see that A is 1-s-strongly stationary in P κx x, fix sets S 0 , S 1 ⊆ P κ X that are both ≺-cofinal in P κx x.Let T be the set of y ∈ P κx x such that S 0 and S 1 are both ≺-cofinal in P κy y.We claim that T is ≺-cofinal in P κx x.To see this, fix an arbitrary y 0 ∈ P κx x.Define a continuous, ≺-increasing sequence y η | η < κ x in P κx x as follows.The set y 0 is already fixed.Given y η , find z 0 η ∈ S 0 and z 1 η ∈ S 1 such that, for all i < 2, we have The set of η < κ x for which κ yη = η is club in κ x , so, since κ x is weakly Mahlo, we can fix some regular cardinal η < κ x such that κ yη = η.A now-familiar argument then shows that S 0 and S 1 are both ≺-cofinal in y η , and hence y η ∈ T .
Since A is 1-strongly stationary in P κx x, we can find w ∈ A such that T is ≺cofinal in P κw w.It follows immediately that S 0 and S 1 are both ≺-cofinal in P κw w; therefore, A is 1-s-strongly stationary in P κx x.
For the "moreover" clause, assume that κ x is strongly inaccessible and A is 1strongly stationary in P κx x.The fact that κ x is Mahlo follows from the previous paragraphs.To show that A is strongly stationary in P κx x, suppose C is a weak club subset of P κx x.Since A is 1-strongly stationary there is some y ∈ A such that C is ≺-cofinal in P κy y.Since C is weakly closed we have y ∈ A ∩ C.

3.3.
The τ 1 topology on P κ X.We now discuss the first derived topology τ 1 on P κ X. Recall that this is the topology generated by Remark 3.11.By definition B 1 is a subbase for the first derived topology τ 1 on P κ X, but it is not clear whether it is a base for τ 1 (essentially because of Proposition 3.4).Recall that the subbase for the first derived topology on an ordinal δ is always a base for that topology (see [2]).This difference seems not to create too much difficulty so we proceed with our definition as is, but in Section 3.6 we show that, if we pass to a certain club subset C of P κ X, then the natural restriction of B 1 to C is a base for the subspace topology on C induced by τ 1 .
We will need the following lemma.Lemma 3.12.Fix x ∈ P κ X, and suppose that A is 1-s-strongly stationary in P κx x and A 0 , . . ., A n−1 are all 0-s-strongly stationary (i.e.≺-cofinal) in P κx x, where Proof.First let us use a straightforward inductive argument on n ≥ 2 to show that whenever A 0 , . . ., A n−1 are 0-strongly stationary in P κx x, the set ∩ A is 0-strongly stationary in P κx x.Suppose A 0 and A 1 are ≺-cofinal in P κx x and note that κ x must be a limit cardinal.To show that d Since A is 1-s-strongly stationary in P κx x there is an a ∈ A ∩ P κx x such that A 0 ∩ (y, x) and A 1 ∩ (y, x) are both ≺-cofinal in P κa a, and hence y < a. Therefore a ∈ d 0 (A 0 ) ∩ d 0 (A 1 ) ∩ A ∩ (y, x).Now suppose the result holds for n, and suppose A 0 , . . ., A n−1 , A n are all ≺-cofinal in P κx x.By our inductive hypothesis, Now we prove the statement of the lemma.Fix sets A 0 , . . ., A n−1 ⊆ P κ X that are ≺-cofinal in P κx x.To show that d 0 (A 0 ) ∩ • • • ∩ d 0 (A n−1 ) ∩ A is 1-s-strongly stationary in P κx x, fix sets S and T that are ≺-cofinal in P κx x.By the previous paragraph, it follows, that the set x and hence there is some Corollary 3.13.Suppose P κx x is 1-s-strongly stationary.Then a set A is 1-sstrongly stationary in P κx x if and only if for all sets C which are 0-s-weak club in Proof.Suppose A is 1-s-strongly stationary in P κx x and C is 0-s-weak club in P κx x.Then d 0 (C) ∩ P κx x ⊆ C ∩ P κx x and by Lemma 3.12, d 0 (C) ∩ A is 1-sstrongly stationary in P κx x.Thus A ∩ C ∩ P κx x = ∅.Conversely, assume that A∩C ∩P κx x = ∅ whenever C is 0-s-weak club in P κx x.Fix sets S and T that are 0s-strongly stationary in P κx x.Then d 0 (S) ∩ d 0 (T ) is 0-s-weak club in P κx x because d 0 (S) ∩ d 0 (T ) ∩ P κx x is 1-s-strongly stationary and hence 0-s-strongly stationary in P κx x by Lemma 3.12, and d 0 (S) as a consequence of the fact that d 0 is the limit point operator of the space (P κ X, τ 0 ).Proposition 3.14.If A ⊆ P κ X then Proof.Suppose A is not 1-s-strongly stationary in P κx x.If κ x is a successor cardinal then x is isolated in (P κ X, τ 1 ) by Corollary 3.2 and hence x / ∈ d 1 (A).Suppose κ x is a limit cardinal.Then there are sets S and T which are 0-strongly stationary in P κx x such that d 0 (S) ∩ d 0 (T ) ∩ A ∩ P κx x = ∅.Then it follows that d 0 (S) ∩ d 0 (T ) ∩ (0, x] is an open neighborhood of x in the τ 1 topology that does not intersect A in some point other than x.Hence x / ∈ d 1 (A).Conversely, suppose A is 1-s-strongly stationary in . Then the sets A 0 , . . ., A n−1 are all ≺-cofinal in P κx x, and by Lemma 3.12, the set Corollary 3.15.A point x ∈ P κ X is not isolated in (P κ X, τ 1 ) if and only if P κx x is 1-s-strongly stationary in P κx x.
3.4.The τ ξ topology on P κ X for ξ ≥ 2. We now move to the general setting.Let us first characterize limit points of sets in the spaces (P κ X, τ ξ ) in terms of ξ-s-strong stationarity.Theorem 3.16.For all ξ < κ the following hold.
(1) ξ We have (2) ξ For all x ∈ P κ X, a set A is ξ + 1-s-strongly stationary in P κx x if and only if for all ζ ≤ ξ and every pair S, T of subsets of P κx x that are ζ-sstrongly stationary in P κx x, we have ) ξ For all x ∈ P κ X, if A is ξ-s-strongly stationary in P κx x and A i is ζ is-strongly stationary in P κx x for some ζ i < ξ and all i < n, then ) is ξ-s-strongly stationary in P κx x.
Proof.We have already established that (1) ξ , (2) ξ and (3) ξ hold for ξ ≤ 1.Given these base cases, the fact that (1), ( 2) and ( 3) hold for all ξ < κ can be established by simultaneous induction using an argument which is essentially identical to that of [2, Proposition 2.10].For the reader's convenience, we include the argument here.
But, by Lemme 3.5, One can show that if (1) ≤ξ , (2) ≤ξ and (3) ≤ξ hold then, by induction on n, (3) ξ+1 must also hold.For the reader's convenience we provide a proof that (3) ξ+1 holds for n = 1, the remaining case is the same as [2, Proposition 2.10].Suppose n = 1.To prove that A ∩ d ζ0 (A 0 ) is ξ + 1-s-strongly stationary in P κx x, fix sets S and T that are η-s-strongly stationary in P κx x for some η ≤ ξ.By (1) ≤ξ , it will suffice to show that Let us prove that if (1) ≤ξ , (2) ≤ξ and (3) ≤ξ+1 hold then (1) ξ+1 holds (this argument is similar to that of Proposition 3.14).Suppose A is not ξ + 1-s-strongly stationary in P κx x.Then by (1) ≤ξ , there there are sets S and T which are ζ-s-strongly is an open neighborhood of x in the τ ξ+1 topology that does not intersect A in some point other than x.Conversly, suppose A is ξ + 1-s-strongly stationary in P κx x.To show that x ∈ d ξ+1 (A), let U be an arbitrary basic open neighborhood of x in the τ ξ+1 topology.By Lemma 3.6, we can assume that U is of the form where I ∈ B 0 , n < ω, ζ < ξ + 1 and A i ⊆ P κ X for i < n.Since x ∈ U it follows from (1) ζ that each A i is ζ-s-strongly stationary in P κx x, and thus by (3) ) is ξ + 1-s-strongly stationary in P κx x, and thus U intersects A in some point other than x.
Corollary 3.17.Suppose P κx x is ξ-s-strongly stationary where ξ ≤ κ x and A is ζ-s-strongly stationary in P κx x for some ζ < ξ.Then, for all Corollary 3.18.Suppose P κx x is ξ + 1-s-strongly stationary.Then a set A is ξ + 1-s-strongly stationary in P κx x if and only if A ∩ C = ∅ for all sets C ⊆ P κx x which are ξ-s-weak club in P κx x.Thus the filter generated by the ξ-s-weak club subsets of P κx x is the filter dual to NS ξ+1 κx,x .Proof.Suppose A is ξ + 1-s-strongly stationary in P κx x and C is ξ-s-weak club in P κx x.By Theorem 3.16 (1), it follows that d ξ (C) ⊆ C and by Theorem 3.16 (3) we see that d ξ (C) ∩ A is ξ + 1-s-strongly stationary in P κx x and thus C ∩ A ∩ P κx x = ∅.
Conversely, suppose A∩C = ∅ whenever C is a ξ-s-weak club subset of P κx x.To show that A is ξ +1-s-strongly stationary in P κx x, suppose S and T are ζ-s-strongly stationary in P κx x for some ζ ≤ ξ.Then the set d ζ (S) ∩ d ζ (T ) is ξ-s-weak club in P κx x because it is ξ-s-strongly stationary in P κx x by Theorem 3. 16(3) and it is ξ-s-closed in P κx x since and hence A is ξ + 1-s-strongly stationary in P κx x as desired.
Corollary 3.19.Suppose that x ∈ P κ X and ξ ≤ κ x .Then x is not isolated in (P κ X, τ ξ ) if and only if P κx x is ξ-s-strongly stationary.
Proof.For the forward direction, suppose that P κx x is not ξ-s-strongly stationary.Then there is ζ < ξ and sets S, T ⊆ P κx x such that S and T are both ζ-s-strongly stationary in P κx x but there is not y ≺ x such that S and T are both ζ-s-strongly stationary in P κy y.Then, by Theorem 3.16(1), we have For the converse, suppose that P κx x is ξ-s-strongly stationary, and fix an interval I ∈ B 0 , an n < ω, ordinals ξ 0 , . . ., ξ n−1 < ξ, and sets A 0 , . . ., A n−1 ⊆ P κx x such that x  Proposition 3.21.For x ∈ P κ X and ξ ≤ κ x , the set P κx x is ξ-s-strongly stationary if and only if NS ξ κx,x is a nontrivial ideal.
Proof.Suppose P κx x is 0-s-strongly stationary.Then NS 0 κx,x is the ideal I κx,x consisting of all subsets A of P κx x such that there is some y ∈ P κx x with A∩(y, x) = ∅.Clearly this is a nontrivial ideal since P κx x / ∈ I κx,x .Now suppose ξ > 0. Let us show that NS ξ κx,x is an ideal.Suppose A and B are both not ξ-s-strongly stationary in P κx x.By Corollary 3.20, there are sets strongly stationary in P κx x by Theorem 3.16(3) and furthermore Theorem 3.22.Suppose that 0 < ξ < κ.Then the following are equivalent: (1) B ξ is a base for τ ξ ; (2) for every ζ ≤ ξ, every x ∈ P κ X, and every We can therefore fix an η < ζ and sets S, T ⊆ P κx x such that S and T are both η-s-strongly stationary in P κx x but there is no y ∈ P κx x such that S and T are both η-s-strongly stationary in P κx x.Then we have d η (S) ∩ d η (T ) = {x}, and hence {x} ∈ τ ξ .To show that (1) fails, it thus suffices to show that {x} / ∈ B ξ .Since P κx x is 1-strongly stationary, it follows that κ x is a limit cardinal, and hence {x} / ∈ B 0 .Now suppose that B ⊆ P κx x, ξ 0 < ξ, and x ∈ d ξ0 (B).Since P κx x is not ζ-s-strongly stationary, it follows that ξ 0 < ζ and B is ξ 0 -s-stationary in P κx x.By minimality of ζ, B is ξ 0 -stationary in P κx x, so, since P κx x is ζ-strongly stationary, there is y ∈ P κx x such that B is ξ 0 -strongly stationary in P κy y.Again by minimality of ζ, B is ξ 0 -s-strongly stationary in P κy y, so y ∈ d ξ0 (B).It follows that {x} / ∈ B ξ .For the backward direction, suppose that (2) holds, and fix x ∈ P κ X, I ∈ B 0 , 0 < n < ω, ordinals ξ 0 , . . ., ξ n−1 < ξ, and sets A 0 , . . ., A n−1 ⊆ P κx x such that Let ζ := max{ξ 0 , . . ., ξ n−1 } < ξ.It follows that P κx is ζ-s-strongly stationary.If P κx x is not (ζ + 1)-strongly stationary, then there is A ⊆ P κx x such that d ζ (A) = {x}.We can therefore assume that P κx x is (ζ + 1)-strongly stationary and hence, by ( 2), (ζ + 1)-s-strongly stationary.But then it follows that

Consequences of Π 1
ξ -indescribability.In this section we establish the consistency of the ξ-s-strong stationarity of P κx x, for ξ ≤ κ x , using a two-cardinal version of transfinite indescribability.
The classical notion of Π m n -indescribability studied by Levy [22] was generalized to the two-cardinal setting in a set of handwritten notes by Baumgartner (see [9,Section 4]).More recently, various transfinite generalizations of classical Π 1 n -indescribability, involving certain infinitary formulas have been studied in the cardinal context [2,3,4,11,13,15] and in the two-cardinal context [12].
Let us review the definition of Π 1 ξ -indescribability in the two-cardinal context used in [12].For the reader's convenience, we review the notion of Π 1 ξ formula introduced in [2].Recall that a formula of second-order logic is Π 1 0 , or equivalently Σ 1 0 , if it does not have any second-order quantifiers, but it may have finitely-many firstorder quantifiers and finitely-many first and second-order free variables.Bagaria inductively defined the notion of Π 1 ξ formula for any ordinal ξ as follows.
where ϕ is Π 1 ξ , and a formula is where ϕ ζ is Π 1 ζ for all ζ < ξ and the infinite conjunction has only finitely-many free second-order variables.We say that a formula is Σ 1 where ϕ ζ is Σ 1 ζ for all ζ < ξ and the infinite disjunction has only finitely-many free second-order variables.
Corollary 3.29.For ξ < κ, if there is an x ∈ P κ X such that P κx x is Π 1 ξindescribable then the τ ξ+1 -topology on P κ X is not discrete.Proposition 3.30.Suppose P κ X is Π 1 1 -indescribable.Then a set A ⊆ P κ X is 2-sstrongly stationary in P κ X if and only if for every pair S, T of strongly stationary subsets of P κ X there is an x ∈ A such that x ∩ κ = κ x is a Mahlo cardinal and the sets S and T are both strongly stationary in P κx x.
Proof.Suppose A is 2-s-strongly stationary in P κ X. Fix sets S and T that are strongly stationary in P κ X.The fact that κ is Mahlo and the sets S and T are strongly stationary in P κ X can be expressed by a Π 1  1 sentence: (V κ (κ, X), ∈, P κ X, S, T ) |= ϕ.
The set Thus C is, in particular, strongly stationary in P κ X and so by Lemma 3.10 we see that C is 1-s-strongly stationary in P κ X.Since A is 2-s-strongly stationary in P κ X, there is an x ∈ A ∩ C and it follows that κ x is Mahlo and the sets S and T are strongly stationary in P κx x.
Conversely, to show that A is 2-s-strongly stationary in P κ X, fix sets Q and R that are 1-s-strongly stationary in P κ X.By Lemma 3.10, Q and R are strongly stationary in P κ X.Thus, by assumption, there is an x ∈ A such that x ∩ κ = κ x is Mahlo and the sets Q and R are both strongly stationary in P κx x.By Lemma 3.10, Q and R are both 1-s-strongly stationary in P κx x.Hence A is 2-s-strongly stationary in P κ X. Proposition 3.31.For x ∈ P κ X with x ∩ κ = κ x , if P κx x is Π 1 ξ -indescribable where ξ < κ x , then the ideal NS ξ+1 κx,x (see Definition 3.9) is strongly normal.Proof.Suppose C z ∈ (NS ξ+1 κx,x ) * for z ∈ P κx X.Without loss of generality, by Corollary 3.18, we may assume that each C z is ξ-s-weak club in P κx x.
Since each C z is in the filter Π 1 ξ (κ x , x) * and Π 1 ξ (κ x , x) is strongly normal, it follows that the set * and thus C is ξ + 1-s-strongly stationary in P κx x by Corollary 3.28.By Theorem 3.16 (2), it follows that d ξ (C) is ξ-s-strongly stationary in P κx x, and since d ξ is the Cantor derivative of the space (P κ X, τ ξ ), it follows that Since each C z is ξ-s-weak club in P κx x, it follows that d ξ (C z ) ⊆ C z and thus 3.6.Variations.In this subsection, we investigate a couple of variations on the sequence of derived topologies considered above.First, we show that by restricting our attention to a certain natural club subset of P κ X, certain questions about the resulting spaces become more tractable.Let P ′ κ X be the set of x ∈ P κ X for which κ x = x ∩ κ.Similarly, if x ∈ P ′ κ X, then P ′ κx x = P ′ κ X ∩ P κx x.If κ is weakly inaccessible, then P ′ κ X is evidently a club, and hence a weak club, in P κ X.It follows that, if ξ < κ, x ∈ P κ X, and κ x is weakly inaccessible, then For each ξ < κ, let τ ′ ξ be the subspace topology on P ′ κ X induced by τ ξ , and let Proposition 3.32.Suppose that x ∈ P ′ κ X.Then the following are equivalent: (1) κ x is weakly inaccessible; (2) x is not isolated in (P ′ κ X, τ ′ 0 ).Proof.If κ x is weakly inaccessible and y ≺ x, with y ∈ P κ X, then, letting λ be the least cardinal with |y| < λ, we have y ∪ λ ∈ (y, x] ∩ P ′ κ X.The implication (1) =⇒ (2) follows immediately.
For the converse, suppose first that κ x = λ + is a successor cardinal, and let y ≺ x be such that |y| = λ.Then (y, x] = {x}, so x is isolated in τ 0 , and hence also in τ ′ 0 .Suppose next that κ x is singular, and let y ⊆ κ x be a cofinal subset such that |y| = cf(κ x ).Then (y, x] ∩ P ′ κ X = {x}, so x is isolated in τ ′ 0 .
Using this proposition, we can establish the following characterization of when B ′ ξ forms a base for τ ′ ξ .Since the proof is essentially the same as that of Theorem 3.22, we leave it to the reader.Theorem 3.33.Suppose that 0 < ξ < κ.Then the following are equivalent: (1) B ′ ξ is a base for τ ′ ξ ; (2) for every ζ ≤ ξ, every x ∈ P ′ κ X for which κ x is weakly inaccessible, and every A ⊆ P κ X, if A is ζ-strongly stationary in P κx x, then A is ζ-s-strongly stationary in P κx x.
We saw above that the topology (P κ X, τ 1 ) can be characterized by specifying that, if x ∈ P κ X and A ⊆ P κ X, then x is a limit point of A if and only if A is strongly 1-s-stationary in P κx x.By Proposition 3.10, if κ x is regular, then this is equivalent to A being 1-strongly stationary in P κx x, and if κ x is Mahlo, it is in turn equivalent to A being strongly stationary in P κx x.One can ask if there is a variant on this topology in which limit points are characterized by stationarity in the sense of [21] (recall the discussion at the end of Section 2).We now show that the answer is positive as long as κ is weakly inaccessible and one only requires this of x ∈ P κ X for which κ x is weakly inaccessible.We first establish the following proposition.less than λ + in the generic ultrapower obtained by forcing with any normal ideal I on Z ⊆ P (λ).We recursively define f α | α < λ + as follows.For α < λ we let where b λ,α : λ → α is a bijection.Let us note that if we take Z = λ, then each f α represents the ordinal α in any generic ultrapower obtained by forcing with a normal ideal on λ.Whereas, in the two-cardinal setting, if we take Z = P κ λ, the function f α represents α in any generic ultrapower obtained by forcing with a normal ideal on P κ λ.
Let us review some basic definitions concerning ineffable and Ramsey operators on cardinals.For S ⊆ κ, we say that S = S α | α ∈ S is an S-list if S α ⊆ α for all α ∈ S. Given an S-list S, a set H ⊆ S is said to be homogeneous for S if whenever α, β ∈ H with α < β we have S α = S β ∩ α.If I is an ideal on κ, we define another ideal I(I) such that for S ⊆ κ we have S ∈ I(I) + if and only if for every S-list S = S α | α ∈ S there is a set H ∈ P (S) ∩ I + which is homogeneous for S. We say that κ is almost ineffable if κ ∈ I([κ] <κ ) + and κ is ineffable if κ ∈ I(NS κ ) + .The function I is referred to as the ineffable operator on κ.
Recall that for a cardinal κ and a set S ⊆ κ, a function f If I is an ideal on a cardinal κ, we define another ideal R(I) such that for S ⊆ κ we have S ∈ R(I) + if and only if for every function f : [κ] <ω → κ that is regressive on S, there is a set H ∈ P (S) ∩ I + which is homogeneous for f .We say that a set S ⊆ κ is Ramsey in κ if S ∈ R([κ] <κ ) + .Let us note that the definition of Ramsey set and, more generally, the definition of R(I) given above are standard and have many equivalent formulations (see [13,  We say that a set S ⊆ κ is γ-almost ineffable if S ∈ I γ ([κ] <κ ) and we say that S ⊆ κ is γ-Ramsey in κ if S ∈ R γ ([κ] <κ ) + .So, for example, a set S ⊆ κ is 1-Ramsey in κ if and only if it is Ramsey in κ, and S is 2-Ramsey in κ if and only if for every function f : [κ] <ω → κ that is regressive on S there is a set H that is Ramsey in κ and homogeneous for f .

4.2.
New results on two-cardinal Ramseyness.Let us now discuss two-cardinal versions of the ineffable and Ramsey operator, which are defined using the strong subset ordering ≺.Suppose κ is a cardinal and X is a set of ordinals with κ ⊆ X.
For S ⊆ P κ X, we say that S = S x | x ∈ P κ X is an (S, ≺)-list if S x ⊆ P κx x for all x ∈ S. Given an (S, ≺)-list, a set H ⊆ S is said to be homogeneous for S if whenever x, y ∈ H with x ≺ y we have S x = S y ∩ P κx x.If I is an ideal on P κ X, we define another ideal I ≺ (I) such that for S ⊆ P κ X we have S ∈ I ≺ (I) + if and only if for every (S, ≺)-list S there is a set H ∈ P (S) ∩ I + which is homogeneous for S. We say that P κ X is strongly ineffable if P κ X ∈ I ≺ (NSS κ,X ) + and almost strongly ineffable if P κ X ∈ I ≺ (I κ,X ) + .Here I κ,X is the ideal on P κ X consisting of all subsets of P κ X which are not ≺-cofinal in P κ X.
Let [S] <ω ≺ be the collection of all tuples x = (x 0 , . . ., x n−1 ) ∈ S n such that n < ω and constant for all n < ω.For S ⊆ P κ X, let S ∈ R ≺ (I) + if and only if for every function f : [P κ X] <ω → P κ X that is ≺-regressive on S, there is a set H ∈ P (S) ∩ I + which is homogeneous for f .We say that P κ X is strongly Ramsey The first author and Philip White [16] showed that many results from the literature [5,6,13,15,17] on the ineffable operator I and the Ramsey operator R, and their relationship with indescribability, can be extended to I ≺ and R ≺ .For example, by iterating the ideal operators I ≺ and R ≺ , one obtains hierarchies in the two-cardinal setting which are analogous to the classical ineffable and Ramsey hierarchies.One question left open by [16] is that which is analogous to Question 4.4 for the two-cardinal context.For example, if P κ X ∈ R 2 ≺ (I κ,X ) + , does it follow that the set The proof of Theorem 4.7 generalizes in a straight-forward way to establish the following.

Questions and ideas
Let us formulate a few open questions relavant the topics of this article.For this section, let us assume κ is some regular uncountable cardinal and X ⊇ κ is a set of ordinals.First, we consider the following questions regarding the consistency strength of various principles considered above.Question 5.1.What is the consistency strength of "whenever S ⊆ P κ X is strongly stationary there is some x ∈ P κ X for which S∩P κx x is strongly stationary in P κx x"?Is this similar to the situation for cardinals?Is the strength of this kind of reflection of strong stationary sets strictly between the "great Mahloness" of P κ X and the Π 1  1 -indescribability of P κ X? Question 5.2.What is the consistency strength of the 2-s-strong stationarity of P κ X? What is the consistency strength of the hypothesis that whenever S and T are strongly stationary in P κ X there is some x ∈ P κ X such that S and T are both strongly stationary in P κx x?
The following questions regarding separation of various properties considered in this article remain open.Question 5.3.Can we separate reflection of strongly stationary sets from pairwise simultaneous reflection of strongly stationary sets?In other words, is it consistent that whenever S is strongly stationary in P κ X there is some x ∈ P κ X such that S is strongly stationary in P κx x, but at the same time, pairwise reflection fails in the sense that there exists a pair S, T of strongly stationary subsets of P κ X such that for every x ∈ P κ X both S and T are not strongly stationary in P κx x?
It is conceivable that some two-cardinal (κ)-like principle could be used to address Questions 5.3.For example, (κ) implies that every stationary subset of κ can be partitioned into two disjoint stationary sets that do not simultaneously reflect (see [20,Theorem 2.1] as well as [14,Theorem 7.1] and [8,Theorem 3.50] for generalizations).Question 5.4.Is some two-cardinal (κ)-like principle formulated using weak clubs (defined in Section 2) consistent?Does it deny pairwise simultaneous reflection of strongly stationary subsets of P κ X?
It is also natural to ask whether the various reflection properties introduced here can be separated from the large cardinal notions that imply them.Question 5.5.Can we separate ξ + 1-strong stationarity or ξ + 1-s-strong stationarity in P κ X from (1) Π 1 ξ -indescribability in P κ X similar to what was done in [3]; or (2) Π 1  1 -indescribability in P κ X similar to what was done in [7]?In [3], it was shown that consistently NS ξ+1 κ can be non-trivial and κ is not Π 1 ξ -indescribable.In [7, Definition 0.7], a normal version of the ideal NS ξ κ was introduced, NS ξ,d κ .It was shown that consistently, NS ξ,d κ can be non-trivial for all ξ < ω while κ is not even Π 1 1 -indescribable.Question 5.6.Is it consistent that κ ∈ I(Π 1 ξ (κ)) and κ ∈ I(NS ξ+1 κ ).Is it consistent that κ ∈ I(Π 1 1 (κ)) and κ / ∈ I(NS ξ,d κ ) for all ξ < ω? Finally, let us consider some questions that arise by considering Proposition 3.8 and [11].Bagaria noticed that, using the definitions of [2], no ordinal α is α + 1stationary (see the discussion after Definition 2.6 in [2]) and no cardinal κ is Π 1 κindescribable (see the discussion after Definition 4.2 in [2]).The first author showed that Bagaria's definitions of ξ-s-stationarity and derived topologies τ ξ | ξ < δ on an ordinal δ, can be modified in a natural way so that a regular cardinal µ can cary a longer sequence of derived topologies τ ξ | ξ < µ + , such that, for each ξ < µ there is a club C ξ in δ such that α ∈ C ξ is not isolated in the τ ξ topology if and only if α is f µ ξ (α)-s-stationary3 (see [11,Theorem 6.15]).The first author also generlized Bagaria's notion of Π 1 ξ -indescribability so that a cardinal κ can be Π 1 ξ -indescribable for all ξ < κ + , and that the Π 1 ξ -indescribability of κ implies the ξ + 1-s-stationarity of κ for all ξ < κ + (see [11,Proposition 6.18]).It is natural to ask whether similar techniques can be used to generalize the results in Section 3.2 of the present article.For example, can one modify the definition of ξ-strong stationarity so that Proposition 3.8 can fail for the modified notion?Question 5.7.Can one use canonical functions to modify the definition of ξ-sstrong stationarity so that it is possible for x ∈ P κ X to be ξ-strongly stationary or ξ-s-strongly stationary for some ξ > κ x ?Question 5.8.Can the definitions of two-cardinal Π 1 ξ -indescribability (Definition 3.24), ξ + 1-s-strong stationarity (Definition 3.7), and the two-cardinal derived topologies (see Section 3.2) be modified using canonical functions so that Corollary 3.28 might generalize to values of ξ for which κ x < ξ < |x| + and Theorem 3.16 might generalize to values of ξ for which κ < ξ < |X| + ?
and by the base case the set d 0 Fix ζ ≤ ξ and a pair S, T of ζ-s-strongly stationary subsets of P κx x.To show that A ∩ d ζ (S) ∩ d ζ (T ) is ζ-sstrongly stationary in P κx x, fix sets A, B that are η-s-strongly stationary in P κx x where η < ζ.Using the fact that (3) holds for ζ, we see that S ∩ d η (A) ∩ d η (B) is ζ-s-strongly stationary in P κx x.Since A is ξ + 1-s-strongly stationary, and applying the fact that (1) ζ holds, we have which implies that d ζ (S) ∩ d ζ (T ) is ζ-s-weak club in P κx x.Conversely, suppose A ∩ d ζ (S) ∩ d ζ (T ) = ∅ whenever S and T are ζ-s-strongly stationary in P κx x for some ζ ≤ ξ +1.Then it easily follows by (1) ≤ ξ that A is ξ +2-s-strongly stationary in P κx x.
Proposition 2.8 and Theorem 2.10] for details).The function R is called the Ramsey operator on κ.For a given ideal I and ideal operator O, such as O ∈ {I, R}, we inductively define new ideals by letting O 0 (I) = I, O α+1 (I) = O(O α (I)) and O α (I) = β<α O β (I).
).Let ζ := max{ζ i | i < n} < ξ.By Corollary 3.17, each of I, d ξ0 (A 0 ), ...,d ξn−1 (A n−1 ) is ζ-s-weak club in P κx x.By Corollary 3.18, U is also ζ-s-weak club in P κx x.In particular, U = {x}; hence, x is not isolated in P κx x.Corollary 3.20.Suppose P κx x is ξ-s-strongly stationary where 0 < ξ ≤ κ x .Then a set A is ξ-s-strongly stationary in P κx x if and only if for all ζ < ξ we have A ∩ C = ∅ for all sets C ⊆ P κx x which are ζ-s-weak club in P κx x.
Proof.Suppose A is ξ-s-strongly stationary in P κx x.Fix ζ < ξ and assume thatC ⊆ P κx x is ζ-s-weak club in P κx x.Since C is ζ-s-strongly stationary in P κx x there is some y ∈ d ζ (C) ∩ A, but since d ζ (C) ⊆ C we have y ∈ C ∩ A. Conversely,suppose that for all ζ < ξ and all C ⊆ P κx x that are ζ-s-weak club in P κx x we have A ∩ C = ∅.To show that A is ξ-s-strongly stationary in P κx x, suppose S and T are ζ-s-strongly stationary in P κx x for some ζ < ξ.Then, since we are assuming that P κx x is ξ-s-strongly stationary, it follows by Theorem 3.16(3) that d ζ (S) ∩ d ζ (T ) is ξ-s-strongly stationary in P κx x.Furthermore, strongly stationary in P κx x.Proof.For the forward direction, suppose that (2) fails, and let ζ, x, and A form a counterexample, with ζ minimal among all such counterexamples.Note that we must have ζ > 0. Claim 3.23.P κx x is not ζ-s-strongly stationary.Proof.Suppose otherwise.We will show that A is in fact ζ-s-strongly stationary, contradicting our choice of A. By Corollary 3.20, it suffices to show that, for all η < ζ and every η-s-weak club C in P κx x, we have A ∩ C = ∅.Fix such η and C. Then C is η-s-strongly stationary in P κx x and hence, by the minimality of ζ, η-strongly stationary in P κx x.Thus, since A is ζ-strongly stationary, there is y ∈ A such that A is η-strongly stationary in P κy y and hence, again by the minimality of ζ, η-s-strongly stationary in P κy y.But then, since C is an η-s-weak club in P κx x, we have y ∈ C ∩ A, as desired.