- Email: [email protected]
On cardinal sequences of LCS spaces
Topology and its Applications 203 (2016) 91–97
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
On cardinal sequences of LCS spaces Juan Carlos Martínez 1 Facultat de Matemàtiques, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain
a r t i c l e
i n f o
Article history: Received 21 July 2014 Accepted 8 March 2015 Available online 11 January 2016 MSC: 54A25 54D30 54G12 03E35 06E05 Keywords: Locally compact scattered space Cardinal sequence Forcing
a b s t r a c t If α is an ordinal, we denote by C(α) the class of all cardinal sequences of length α associated with locally compact scattered spaces (its precise definition is given in Section 1). In this paper, we present a general construction of locally compact scattered spaces with a large top. As consequences of this construction we obtain the following results: (1) If κ is a singular cardinal of cofinality ω, then κκ κω ∈ C(κ + 1). (2) If κ is an inaccessible cardinal, then κκ κκ ∈ C(κ + 1). (3) If GCH holds, then for any infinite cardinal κ we have κκ κcf(κ) ∈ C(κ + 1). Also, we prove that if κ is a singular cardinal of cofinality ω, then for every cardinal λ such that κ < λ ≤ κω we have that κκ λω2 ∈ C(κ + ω2 ). © 2016 Elsevier B.V. All rights reserved.
1. Introduction All spaces in this paper are assumed to be Hausdorff. Recall that a space X is scattered, if every non-empty subspace of X has an isolated point. Then, by an LCS space we mean a locally compact, Hausdorff and scattered space. If X is an LCS space and α is an ordinal, we define the α-th Cantor–Bendixson level of X by Iα (X) = the set of isolated points of X \ {Iβ (X) : β < α}. The height of an LCS space X is defined by ht(X) = the least ordinal α such that Iα (X) = ∅. And the reduced height of X is defined by ht− (X) = the least ordinal α such that Iα (X) is finite. Clearly, one has ht− (X) ≤ ht(X) ≤ ht− (X) + 1. The cardinal sequence of X, in symbols CS(X), is defined as the sequence formed by the cardinalities of the infinite Cantor–Bendixson levels of X, i.e. CS(X) = |Iα (X)| : α < ht− (X).
1
E-mail address: [email protected]. Supported by the Spanish Ministry of Education DGI grant MTM2014-59178-P and by the Catalan DURSI grant 2014SGR437.
http://dx.doi.org/10.1016/j.topol.2015.12.078 0166-8641/© 2016 Elsevier B.V. All rights reserved.
92
J.C. Martínez / Topology and its Applications 203 (2016) 91–97
Many authors have studied the possible sequences of infinite cardinals that can arise as the cardinal sequence of an LCS space. It was proved by Juhász and Weiss that if f = κα : α < ω1 is a sequence of infinite cardinals, then f is the cardinal sequence of an LCS space iff κβ ≤ κω α for every α < β < ω1 (see [7]). The situation becomes, however, more complicated when we consider cardinal sequences of length greater than ω1 . Nevertheless, a characterization under GCH for cardinal sequences of length < ω2 was obtained by Juhász, Soukup and Weiss in [5]. However, no characterization is known for cardinal sequences of length ω2 . We refer the reader to the survey papers [1] and [14] for a wide list of results on cardinal sequences as well as examples and basic facts. In particular, it is known that the notion of an LCS space is forcing indestructible and that results on cardinal sequences for LCS spaces can be directly translated into the context of superatomic Boolean algebras. If α is an ordinal, we put C(α) = {CS(X) : X is an LCS space and ht− (X) = α}. If κ is an infinite cardinal and α is an ordinal, we denote by κα the cardinal sequence κβ : β < α where κβ = κ for β < α. If f and g are sequences of infinite cardinals, we denote by f g the concatenation of f with g. Assume that X is an LCS space. We define the width of X by wd(X) = sup{|Iα (X)| : α < ht(X)}. If κ is an infinite cardinal, we say that X is κ-thin-tall, if κ ≤ wd(X) < |ht(X)|. And if γ is an infinite ordinal, we say that X is a γ-thin-thick space, if CS(X) = κα : α ≤ γ where κα ≤ |γ| for α < γ and κγ ≥ |γ|+ . It is well known that ωα ∈ C(α) for every ordinal α < ω2 (see [6] or [7]) and that it is relatively consistent with ZFC that ωα ∈ C(α) for every ordinal α < ω3 (see [2] and [12]). Also, it was proved by Just in [8] that in the Cohen model, ωω2 ∈ / C(ω2 ) and ωω1 ω2 ∈ / C(ω1 + 1). In addition, it was proved by Baumgartner in [2] that ω1 ω1 ω2 ∈ / C(ω1 + 1) in the Mitchell model. On the other hand, it was shown in [9] that if V = L holds then for every regular cardinal κ, κκ+ ∈ C(κ+ ) and κκ κ+ ∈ C(κ + 1). And recently, as a consequence of the main theorem of [13], we obtained that if κ, λ are specific infinite cardinals with κ < λ and κ regular, then it is consistent that κκ+ λ ∈ C(κ+ + 1). However, for κ a singular cardinal, no result is known on the existence of κ-thin-tall spaces and very little is known on the existence of κ-thin-thick spaces. In relation to these problems, it was proved in [4] that if κ is an uncountable cardinal such that κ<κ = κ then κκ κ+ κ+ ∈ C(κ+ ). Then, by using the refinement of Prikry forcing due to Magidor (see [3, Chapter 36]) and the well-known fact that the notion of an LCS space is forcing indestructible, we obtain as a corollary of this result that if it is consistent that there is a measurable cardinal, then it is consistent that ℵω ℵω ℵ+ ∈ C(ℵ+ ω ℵ+ ω ). ω Then, we shall show here the following general result on the existence of κ-thin-thick spaces: if κ is an infinite cardinal and δ < κ+ is a limit ordinal such that κ
J.C. Martínez / Topology and its Applications 203 (2016) 91–97
93
In particular, ℵω ℵω ℵω ω ∈ C(ℵω + 1), i.e. in ZFC there is an ℵω -thin-thick space. Corollary 2.3. If κ is an inaccessible cardinal, then κκ κκ ∈ C(κ + 1). Corollary 2.4. If GCH holds, then for every infinite cardinal κ we have κκ κcf(κ) ∈ C(κ + 1). The following proposition will be useful in the proof of Theorem 2.1. Proposition 2.5. For every infinite cardinal κ and every ordinal α < κ+ , we have that κα ∈ C(α). Proof. Assume that κ is an infinite cardinal. The case α = 0 is obvious. Then, we show by transfinite induction that for every non-zero ordinal α < κ+ there is a non-compact LCS space Xα whose underlying set is κα (ordinal exponentiation) such that CS(Xα ) = κα , and in such a way that if 0 < β < α < κ+ then Xβ is an open subspace of Xα . We define X1 as the cardinal κ with the discrete topology. If α is a limit ordinal, we define Xα as the topological sum of the spaces Xβ for β < α. Now, assume that α = β + 1 with β ≥ 1. Let Z0 be the one-point compactification of Xβ , and for 0 < ξ < κ let Zξ be a homeomorphic copy of Z0 in such a way that Zμ ∩ Zξ = ∅ for μ < ξ < κ. Then, we define Xα as the topological sum of the spaces Zξ for ξ < κ. Note that since the underlying set of Xβ is κβ , we may assume that the underlying set of Xα is κα . Clearly, Xα is as required. 2 Also, note that if Z is the one-point compactification of a non-compact LCS space X, then Z is a compact scattered space with CS(Z) = CS(X). Proof of Theorem 2.1. In order to prove (a), assume that κ is an infinite cardinal and δ < κ+ is a limit ordinal such that κλ = κ for each non-zero cardinal λ < cf(δ). We put T = κ≤cf(δ) . If s, t ∈ T , we put s t iff s ⊆ t. So, (T, ) is the complete κ-ary tree of height cf(δ) + 1. We set B(x) = {v ∈ T : v x} for x ∈ T and B(u, x) = {v ∈ T : u ≺ v x} for u, x ∈ T with u ≺ x. We write T = {κα : α < cf(δ) is non-limit}. Let δα : α < cf(δ) be an increasing sequence of ordinals such that δ = sup{δα : α < cf(δ)}. By Proposition 2.5, for every non-limit ordinal α < cf(δ) we can consider a compact scattered space Xα with ht(Xα ) = δα + 1, |Iβ (Xα )| = κ for every ordinal β < δα and |Iδα (Xα )| = 1. If s ∈ κα where α < cf(δ) is a non-limit ordinal, we take a space Xs homeomorphic to Xα such that Xs ∩ T = {s} = Iδα (Xs ) and in such a way that if s, t ∈ T with s = t, then the underlying sets of Xs and Xt are disjoint. Then, we define the required space X as follows. The underlying set of X is the set
{Xs : s ∈ T } ∪ (T \ T ).
If x ∈ Xs for some s ∈ T , then a basic neighbourhood of x in X is an open neighbourhood of x in Xs . So, assume that x ∈ κα where α ≤ cf(δ) is a limit ordinal. Then, a (clopen) basic neighbourhood of x in X is a set of the form B(u, x) ∪
{Xs : s ∈ T , u ≺ s ≺ x}
where u ∈ T with u ≺ x. Proceeding by transfinite induction on α ≤ cf(δ), we can check that if x ∈ κα where α ≤ cf(δ) is a limit ordinal, then B(x) ∪ {Xs : s ∈ T , s ≺ x} is a compact neighbourhood of x. Therefore, X is locally compact. To check that X is Hausdorff, assume that x, y ∈ X with x = y. Without loss of generality, we may assume that x ∈ κα and y ∈ κβ where α, β ≤ cf(δ) are limit ordinals and that x, y are incomparable in T . Let z ∈ T be such that z ≺ x, y and there is no z ∈ T with z ≺ z ≺ x, y. Then, we take the disjoint neighbourhoods Ux = B(z, x) ∪ {Xs : s ∈ T , z ≺ s ≺ x} and Uy = B(z, y) ∪ {Xs : s ∈ T , z ≺ s ≺ y}.
94
J.C. Martínez / Topology and its Applications 203 (2016) 91–97
Also, note that X is a scattered space of height δ + 1 with Iδ (X) = κcf(δ) . Then, since cf(δ) ≤ κ, κλ = κ for each non-zero cardinal λ < cf(δ) and CS(Xα ) = κδα for each non-limit ordinal α < cf(δ), it is easy to verify that |Iα (X)| = κ for each α < δ, and hence CS(X) = κδ κcf(δ) . Now, in order to prove (b) assume that κ is an infinite cardinal and δ = γ + 1 < κ+ is a successor ordinal. We put T = κ≤ω and we define on T as above. By Proposition 2.5, there is a compact scattered space Y such that ht(Y ) = γ + 1, |Iβ (Y )| = κ for every β < γ and |Iγ (Y )| = 1. If s ∈ κ<ω , we take a space Xs homeomorphic to Y such that Xs ∩ T = {s} = Iγ (Xs ) and in such a way that if s, t ∈ κ<ω with s = t, then the underlying sets of Xs and Xt are disjoint. Now, we define the required space X as follows. The underlying set of X is the set {Xs : s ∈ κ<ω } ∪ κω . If x ∈ Xs for some s ∈ κ<ω , then a basic neighbourhood of x in X is an open neighbourhood of x in Xs . And if x ∈ κω , a basic neighbourhood of x in X is a set of the form {x} ∪ {Xs : u ≺ s ≺ x} where u ∈ T with u ≺ x. Proceeding as above, we see that X is as required. 2 It was proved in [2, Theorem 3.2] that ω1 ω1 ω2 ∈ / C(ω1 + 1) in the Mitchell model. Recall that for θ, an inaccessible cardinal, Mitchell’s forcing is θ-c.c. and θ is collapsed to become ω2 . Thus, we obtain from [2, Theorem 3.2] and Corollary 2.4 the following result. Corollary 2.6. If it is consistent that there is an inaccessible cardinal, then it is consistent that ω1 ω1 ω2 ∈ / C(ω1 + 1) but κκ κ+ ∈ C(κ + 1) for every cardinal κ > ω1 . Also, it was shown in [13] that for every specific infinite cardinals κ, λ with κ < λ and κ regular and every specific ordinal δ with κ+ ≤ δ < κ++ and cf(δ) = κ+ , it is consistent that κδ λ ∈ C(δ + 1). Then, the construction carried out in Theorem 2.1 permits us to extend this result to ordinals δ whose cofinality is smaller than κ+ , and hence we obtain the following theorem. Theorem 2.7. Assume that GCH holds. If κ, λ are infinite cardinals with κ < λ and κ regular and δ < κ++ is an ordinal, then in some cardinal-preserving generic extension we have that κδ λ ∈ C(δ + 1). Proof. If δ < κ++ with cf(δ) = κ+ , we have that κδ λ ∈ C(δ + 1) by [13, Theorem 1.2]. So, without loss of generality, we may assume that κ+ < δ < κ++ with cf(δ) ≤ κ. By the main result of [11], there is a κ-closed and κ+ -c.c. partial order P of size κ+ that forces the existence of an LCS space of height δ and width κ. Note that P preserves GCH (see [10, Lemma 5.12 and Theorem 6.14]). Let G be P-generic and let N = V [G]. Suppose that δ is a limit ordinal. Since N satisfies GCH, there is a forcing extension H of N that preserves cofinalities such that in H, 2θ = θ+ for every cardinal θ < cf(δ) and 2cf(δ) ≥ λ (see [10, Theorems 6.14, 6.16 and 6.17]). Now, in H we consider the tree T = 2≤cf(δ) where for s, t ∈ T , s t iff s ⊆ t. Let δα : α < cf(δ) be an increasing sequence of ordinals cofinal in δ. For every non-limit ordinal α < cf(δ) let Zα be a compact scattered space with ht(Zα ) = δα + 1, |Iβ (Zα )| = κ for every ordinal β < δα and |Iδα (Zα )| = 1. Then, as cf(δ) ≤ κ we can proceed as in the proof of part (a) of Theorem 2.1 and show that there is an LCS space X with CS(X) = κδ 2cf(δ) . Now, as λ ≤ 2cf(δ) , we can take an open subspace Z of X such that ht(Z) = ht(X), Iα (Z) = Iα (X) for every α < δ and |Iδ (Z)| = λ. Clearly, Z is locally compact and scattered. So, we have that κδ λ ∈ C(δ + 1). And if δ is a successor ordinal, we consider a cardinal-preserving forcing extension H of N such that (2ω ≥ λ)H , and then we use the argument given in the proof of part (b) of Theorem 2.1. 2
J.C. Martínez / Topology and its Applications 203 (2016) 91–97
95
3. A construction for singular cardinals In this section, we shall show the following result. Theorem 3.1. If κ is a singular cardinal of cofinality ω, then for every cardinal λ such that κ < λ ≤ κω we have that κκ λω2 ∈ C(κ + ω2 ). The following notion will be used in the proof of Theorem 3.1. Definition 3.2. An LCS space X is special, if ht(X) = ht− (X) and for every point x ∈ X there is a compact open neighbourhood U (x) of x such that the following hold: (1) For every x, y ∈ X, U (x) ⊆ U (y) or U (y) ⊆ U (x) or U (x) ∩ U (y) = ∅. (2) If x, y ∈ Iα (X) for some α < ht(X) with x = y, then U (x) ∩ U (y) = ∅. (3) If x ∈ Iα (X) and α < β < ht(X), then there is a y ∈ Iβ (X) such that U (x) ⊆ U (y). By using the argument given in the proof of Proposition 2.5, we can show the following lemma. Lemma 3.3. For every infinite cardinal κ and every non-zero ordinal α < κ+ , there is a special LCS space X such that CS(X) = κα . Proof of Theorem 3.1. Assume that κ is a singular cardinal of cofinality ω. Let κn : n < ω be a strictly increasing sequence of infinite cardinals cofinal in κ. Let λ be a cardinal with κ < λ ≤ κω . We construct an LCS space X of height κ + ω2 such that |Iα (X)| = κ for α < κ and |Iα (X)| = λ for κ ≤ α < κ + ω2 . By Lemma 3.3, we can consider a special LCS space Z0 such that CS(Z0 ) = κκ . For 0 < n < ω we take a homeomorphic copy Zn of Z0 such that Zm ∩ Zn = ∅ for m < n < ω and Zn ∩ ({α} × λ) = ∅ for n < ω and κ ≤ α < κ + ω2 . Then, the underlying set of the required space X will be a set of the form Z ∪ {{α} × λ : κ ≤ α < κ + ω2 } where Z ⊆ {Zn : n ∈ ω}. We will have that Z = {Iα (X) : α < κ} and Iα (X) = {α} × λ for κ ≤ α < κ + ω2 . Fix n < ω. Since Zn is special, for every ordinal α < κ we can pick a point zn,α ∈ Iα (Zn ) in such a way that U (zn,α ) ∩ {U (zn,β ) : β < α} = ∅. Also as λ ≤ κω , by means of a standard combinatorial argument, we can construct a sequence gξ : ξ < λ of functions in ω κ such that gξ (n) > κn for ξ < λ and n < ω and in such a way that the set {n ∈ ω : gμ (n) = gξ (n)} is finite for μ < ξ < λ. Now, in order to define the required space X, first we fix an enumeration without repetitions {yξ : ξ < λ} of {{α} × λ : κ ≤ α < κ + ω2 }. Proceeding by transfinite induction on ξ < λ we can construct an LCS space Yξ such that the following conditions hold: (1) (2) (3) (4)
yξ ∈ Yξ . ht(Yξ ) = o.t.(δ \ κ) + 2 where δ is the ordinal such that yξ ∈ {δ} × λ. For every ordinal ζ < ht(Yξ ), Iζ (Yξ ) is a countable subset of {κ + ζ} × λ. If μ < ξ < λ, then Yμ ∩ Yξ = ∅.
Clearly, it follows from condition (3) that Yξ ∩ {Zn : n ∈ ω} = ∅. Note that the construction of Yξ can be carried out, because for every ordinal α < ω2 we have that ωα ∈ C(α) (see [6]).
J.C. Martínez / Topology and its Applications 203 (2016) 91–97
96
Now, for every ξ < λ we define a space Xξ from the space Yξ and the function gξ as follows. First, we put Wn,ξ = U (zn,gξ (n) ) for every n ∈ ω. And we put Wξ = {Wn,ξ : n ∈ ω}. Let {vk : k ∈ ω} be an enumeration of I0 (Yξ ). Consider a partition {an : n ∈ ω} of ω into infinite subsets. The underlying set of Xξ is Wξ ∪ Yξ . If x ∈ Wn,ξ for some n ∈ ω, then a basic neighbourhood of x in Xξ is an open neighbourhood of x in Wn,ξ (with the relative topology of Zn ). For k ∈ ω, a basic neighbourhood of vk in Xξ is a set of the form {vk } ∪ {Wn,ξ : n ∈ ak , n > m} where m ∈ ω. And if x ∈ Yξ \ I0 (Yξ ), then a basic neighbourhood of x in Xξ is a set of the form U ∪ {Wn,ξ : n ∈ ak , vk ∈ U } where U is an open neighbourhood of x in Yξ . Then, we define the space X as follows. Its underlying set is {Xξ : ξ < λ}, and its topology has {U ⊆ X : U is an open subset of Xξ for some ξ < λ} as a base. It is easy to check that X is locally compact and scattered. To verify that X is Hausdorff, assume that x ∈ Xμ and y ∈ Xξ with x = y and μ = ξ. Without loss of generality, we may assume that x ∈ Yμ and y ∈ Yξ . Let k ∈ ω be such that {n ∈ ω : gμ (n) = gξ (n)} ⊆ k. Let Vx = (Ux ∪ Wμ ) \ (U (z0,gμ (0) ) ∪ . . . ∪ U (zk−1,gμ (k−1) )) where Ux is an open neighbourhood of x in Yμ , and let Vy = (Uy ∪ Wξ ) \ (U (z0,gξ (0) ) ∪ . . . ∪ U (zk−1,gξ (k−1) )) where Uy is an open neighbourhood of y in Yξ . Note that by the way we have chosen the points zn,α , we have Vx ∩ Vy = ∅. Also, it is clear that CS(X) = κκ λω2 . Thus, X is as required. 2 On the other hand, by the main result of [12], we know that it is relatively consistent with ZFC that ωα ∈ C(α) for every ordinal α < ω3 . So, by using an argument parallel to the one given in the proof of Theorem 3.1, we obtain the following consistency result (without using large cardinals). Theorem 3.4. It is relatively consistent with ZFC that for every singular cardinal κ of cofinality ω and every cardinal λ such that κ < λ ≤ κω , κκ λω3 ∈ C(κ + ω3 ). In particular, we obtain the following result from Theorem 3.1 and Theorem 3.4. Corollary 3.5. (a) ℵω ℵω ℵ+ ω ω2 ∈ C(ℵω + ω2 ). (b) It is relatively consistent with ZFC that ℵω ℵω ℵ+ ω ω3 ∈ C(ℵω + ω3 ). Acknowledgements I would like to express my gratitude to Lajos Soukup and the anonymous referee for several useful comments and suggestions. References [1] J. Bagaria, Thin-tall spaces and cardinal sequences, in: E. Pearl (Ed.), Open Problems in Topology II, Elsevier, Amsterdam, 2007, pp. 115–124. [2] J.E. Baumgartner, S. Shelah, Remarks on superatomic Boolean algebras, Ann. Pure Appl. Logic 33 (2) (1987) 109–129. [3] T. Jech, Set Theory, the third millennium edition, Springer Monogr. Math., Springer-Verlag, Berlin, 2003. [4] I. Juhász, S. Shelah, L. Soukup, Z. Szentmiklóssy, A tall space with a small bottom, Proc. Am. Math. Soc. 131 (6) (2003) 1907–1916. [5] I. Juhász, L. Soukup, W. Weiss, Cardinal sequences of length < ω2 under GCH, Fundam. Math. 189 (1) (2006) 35–52. [6] I. Juhász, W. Weiss, On thin-tall scattered spaces, Colloq. Math. 40 (1) (1978/79) 63–68. [7] I. Juhász, W. Weiss, Cardinal sequences, Ann. Pure Appl. Logic 144 (1–3) (2006) 96–106. [8] W. Just, Two consistency results concerning thin-tall Boolean algebras, Algebra Univers. 20 (2) (1985) 135–142. [9] P. Koepke, J.C. Martínez, Superatomic Boolean algebras constructed from morasses, J. Symb. Log. 60 (3) (1995) 940–951. [10] K. Kunen, Set Theory, North-Holland, Amsterdam, 1980.
J.C. Martínez / Topology and its Applications 203 (2016) 91–97
97
[11] J.C. Martínez, A forcing construction of thin-tall Boolean algebras, Fundam. Math. 159 (2) (1999) 99–113. [12] J.C. Martínez, A consistency result on thin-very tall Boolean algebras, Isr. J. Math. 123 (2001) 273–284. [13] J.C. Martínez, L. Soukup, Superatomic Boolean algebras constructed from strongly unbounded functions, Math. Log. Q. 57 (5) (2011) 456–469. [14] J. Roitman, Superatomic Boolean algebras, in: J.D. Monk, R. Bonnet (Eds.), Handbook of Boolean Algebras, vol. 3, North-Holland, Amsterdam, 1989, pp. 719–740.
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
On cardinal sequences of LCS spaces Juan Carlos Martínez 1 Facultat de Matemàtiques, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain
a r t i c l e
i n f o
Article history: Received 21 July 2014 Accepted 8 March 2015 Available online 11 January 2016 MSC: 54A25 54D30 54G12 03E35 06E05 Keywords: Locally compact scattered space Cardinal sequence Forcing
a b s t r a c t If α is an ordinal, we denote by C(α) the class of all cardinal sequences of length α associated with locally compact scattered spaces (its precise definition is given in Section 1). In this paper, we present a general construction of locally compact scattered spaces with a large top. As consequences of this construction we obtain the following results: (1) If κ is a singular cardinal of cofinality ω, then κκ κω ∈ C(κ + 1). (2) If κ is an inaccessible cardinal, then κκ κκ ∈ C(κ + 1). (3) If GCH holds, then for any infinite cardinal κ we have κκ κcf(κ) ∈ C(κ + 1). Also, we prove that if κ is a singular cardinal of cofinality ω, then for every cardinal λ such that κ < λ ≤ κω we have that κκ λω2 ∈ C(κ + ω2 ). © 2016 Elsevier B.V. All rights reserved.
1. Introduction All spaces in this paper are assumed to be Hausdorff. Recall that a space X is scattered, if every non-empty subspace of X has an isolated point. Then, by an LCS space we mean a locally compact, Hausdorff and scattered space. If X is an LCS space and α is an ordinal, we define the α-th Cantor–Bendixson level of X by Iα (X) = the set of isolated points of X \ {Iβ (X) : β < α}. The height of an LCS space X is defined by ht(X) = the least ordinal α such that Iα (X) = ∅. And the reduced height of X is defined by ht− (X) = the least ordinal α such that Iα (X) is finite. Clearly, one has ht− (X) ≤ ht(X) ≤ ht− (X) + 1. The cardinal sequence of X, in symbols CS(X), is defined as the sequence formed by the cardinalities of the infinite Cantor–Bendixson levels of X, i.e. CS(X) = |Iα (X)| : α < ht− (X).
1
E-mail address: [email protected]. Supported by the Spanish Ministry of Education DGI grant MTM2014-59178-P and by the Catalan DURSI grant 2014SGR437.
http://dx.doi.org/10.1016/j.topol.2015.12.078 0166-8641/© 2016 Elsevier B.V. All rights reserved.
92
J.C. Martínez / Topology and its Applications 203 (2016) 91–97
Many authors have studied the possible sequences of infinite cardinals that can arise as the cardinal sequence of an LCS space. It was proved by Juhász and Weiss that if f = κα : α < ω1 is a sequence of infinite cardinals, then f is the cardinal sequence of an LCS space iff κβ ≤ κω α for every α < β < ω1 (see [7]). The situation becomes, however, more complicated when we consider cardinal sequences of length greater than ω1 . Nevertheless, a characterization under GCH for cardinal sequences of length < ω2 was obtained by Juhász, Soukup and Weiss in [5]. However, no characterization is known for cardinal sequences of length ω2 . We refer the reader to the survey papers [1] and [14] for a wide list of results on cardinal sequences as well as examples and basic facts. In particular, it is known that the notion of an LCS space is forcing indestructible and that results on cardinal sequences for LCS spaces can be directly translated into the context of superatomic Boolean algebras. If α is an ordinal, we put C(α) = {CS(X) : X is an LCS space and ht− (X) = α}. If κ is an infinite cardinal and α is an ordinal, we denote by κα the cardinal sequence κβ : β < α where κβ = κ for β < α. If f and g are sequences of infinite cardinals, we denote by f g the concatenation of f with g. Assume that X is an LCS space. We define the width of X by wd(X) = sup{|Iα (X)| : α < ht(X)}. If κ is an infinite cardinal, we say that X is κ-thin-tall, if κ ≤ wd(X) < |ht(X)|. And if γ is an infinite ordinal, we say that X is a γ-thin-thick space, if CS(X) = κα : α ≤ γ where κα ≤ |γ| for α < γ and κγ ≥ |γ|+ . It is well known that ωα ∈ C(α) for every ordinal α < ω2 (see [6] or [7]) and that it is relatively consistent with ZFC that ωα ∈ C(α) for every ordinal α < ω3 (see [2] and [12]). Also, it was proved by Just in [8] that in the Cohen model, ωω2 ∈ / C(ω2 ) and ωω1 ω2 ∈ / C(ω1 + 1). In addition, it was proved by Baumgartner in [2] that ω1 ω1 ω2 ∈ / C(ω1 + 1) in the Mitchell model. On the other hand, it was shown in [9] that if V = L holds then for every regular cardinal κ, κκ+ ∈ C(κ+ ) and κκ κ+ ∈ C(κ + 1). And recently, as a consequence of the main theorem of [13], we obtained that if κ, λ are specific infinite cardinals with κ < λ and κ regular, then it is consistent that κκ+ λ ∈ C(κ+ + 1). However, for κ a singular cardinal, no result is known on the existence of κ-thin-tall spaces and very little is known on the existence of κ-thin-thick spaces. In relation to these problems, it was proved in [4] that if κ is an uncountable cardinal such that κ<κ = κ then κκ κ+ κ+ ∈ C(κ+ ). Then, by using the refinement of Prikry forcing due to Magidor (see [3, Chapter 36]) and the well-known fact that the notion of an LCS space is forcing indestructible, we obtain as a corollary of this result that if it is consistent that there is a measurable cardinal, then it is consistent that ℵω ℵω ℵ+ ∈ C(ℵ+ ω ℵ+ ω ). ω Then, we shall show here the following general result on the existence of κ-thin-thick spaces: if κ is an infinite cardinal and δ < κ+ is a limit ordinal such that κ
J.C. Martínez / Topology and its Applications 203 (2016) 91–97
93
In particular, ℵω ℵω ℵω ω ∈ C(ℵω + 1), i.e. in ZFC there is an ℵω -thin-thick space. Corollary 2.3. If κ is an inaccessible cardinal, then κκ κκ ∈ C(κ + 1). Corollary 2.4. If GCH holds, then for every infinite cardinal κ we have κκ κcf(κ) ∈ C(κ + 1). The following proposition will be useful in the proof of Theorem 2.1. Proposition 2.5. For every infinite cardinal κ and every ordinal α < κ+ , we have that κα ∈ C(α). Proof. Assume that κ is an infinite cardinal. The case α = 0 is obvious. Then, we show by transfinite induction that for every non-zero ordinal α < κ+ there is a non-compact LCS space Xα whose underlying set is κα (ordinal exponentiation) such that CS(Xα ) = κα , and in such a way that if 0 < β < α < κ+ then Xβ is an open subspace of Xα . We define X1 as the cardinal κ with the discrete topology. If α is a limit ordinal, we define Xα as the topological sum of the spaces Xβ for β < α. Now, assume that α = β + 1 with β ≥ 1. Let Z0 be the one-point compactification of Xβ , and for 0 < ξ < κ let Zξ be a homeomorphic copy of Z0 in such a way that Zμ ∩ Zξ = ∅ for μ < ξ < κ. Then, we define Xα as the topological sum of the spaces Zξ for ξ < κ. Note that since the underlying set of Xβ is κβ , we may assume that the underlying set of Xα is κα . Clearly, Xα is as required. 2 Also, note that if Z is the one-point compactification of a non-compact LCS space X, then Z is a compact scattered space with CS(Z) = CS(X). Proof of Theorem 2.1. In order to prove (a), assume that κ is an infinite cardinal and δ < κ+ is a limit ordinal such that κλ = κ for each non-zero cardinal λ < cf(δ). We put T = κ≤cf(δ) . If s, t ∈ T , we put s t iff s ⊆ t. So, (T, ) is the complete κ-ary tree of height cf(δ) + 1. We set B(x) = {v ∈ T : v x} for x ∈ T and B(u, x) = {v ∈ T : u ≺ v x} for u, x ∈ T with u ≺ x. We write T = {κα : α < cf(δ) is non-limit}. Let δα : α < cf(δ) be an increasing sequence of ordinals such that δ = sup{δα : α < cf(δ)}. By Proposition 2.5, for every non-limit ordinal α < cf(δ) we can consider a compact scattered space Xα with ht(Xα ) = δα + 1, |Iβ (Xα )| = κ for every ordinal β < δα and |Iδα (Xα )| = 1. If s ∈ κα where α < cf(δ) is a non-limit ordinal, we take a space Xs homeomorphic to Xα such that Xs ∩ T = {s} = Iδα (Xs ) and in such a way that if s, t ∈ T with s = t, then the underlying sets of Xs and Xt are disjoint. Then, we define the required space X as follows. The underlying set of X is the set
{Xs : s ∈ T } ∪ (T \ T ).
If x ∈ Xs for some s ∈ T , then a basic neighbourhood of x in X is an open neighbourhood of x in Xs . So, assume that x ∈ κα where α ≤ cf(δ) is a limit ordinal. Then, a (clopen) basic neighbourhood of x in X is a set of the form B(u, x) ∪
{Xs : s ∈ T , u ≺ s ≺ x}
where u ∈ T with u ≺ x. Proceeding by transfinite induction on α ≤ cf(δ), we can check that if x ∈ κα where α ≤ cf(δ) is a limit ordinal, then B(x) ∪ {Xs : s ∈ T , s ≺ x} is a compact neighbourhood of x. Therefore, X is locally compact. To check that X is Hausdorff, assume that x, y ∈ X with x = y. Without loss of generality, we may assume that x ∈ κα and y ∈ κβ where α, β ≤ cf(δ) are limit ordinals and that x, y are incomparable in T . Let z ∈ T be such that z ≺ x, y and there is no z ∈ T with z ≺ z ≺ x, y. Then, we take the disjoint neighbourhoods Ux = B(z, x) ∪ {Xs : s ∈ T , z ≺ s ≺ x} and Uy = B(z, y) ∪ {Xs : s ∈ T , z ≺ s ≺ y}.
94
J.C. Martínez / Topology and its Applications 203 (2016) 91–97
Also, note that X is a scattered space of height δ + 1 with Iδ (X) = κcf(δ) . Then, since cf(δ) ≤ κ, κλ = κ for each non-zero cardinal λ < cf(δ) and CS(Xα ) = κδα for each non-limit ordinal α < cf(δ), it is easy to verify that |Iα (X)| = κ for each α < δ, and hence CS(X) = κδ κcf(δ) . Now, in order to prove (b) assume that κ is an infinite cardinal and δ = γ + 1 < κ+ is a successor ordinal. We put T = κ≤ω and we define on T as above. By Proposition 2.5, there is a compact scattered space Y such that ht(Y ) = γ + 1, |Iβ (Y )| = κ for every β < γ and |Iγ (Y )| = 1. If s ∈ κ<ω , we take a space Xs homeomorphic to Y such that Xs ∩ T = {s} = Iγ (Xs ) and in such a way that if s, t ∈ κ<ω with s = t, then the underlying sets of Xs and Xt are disjoint. Now, we define the required space X as follows. The underlying set of X is the set {Xs : s ∈ κ<ω } ∪ κω . If x ∈ Xs for some s ∈ κ<ω , then a basic neighbourhood of x in X is an open neighbourhood of x in Xs . And if x ∈ κω , a basic neighbourhood of x in X is a set of the form {x} ∪ {Xs : u ≺ s ≺ x} where u ∈ T with u ≺ x. Proceeding as above, we see that X is as required. 2 It was proved in [2, Theorem 3.2] that ω1 ω1 ω2 ∈ / C(ω1 + 1) in the Mitchell model. Recall that for θ, an inaccessible cardinal, Mitchell’s forcing is θ-c.c. and θ is collapsed to become ω2 . Thus, we obtain from [2, Theorem 3.2] and Corollary 2.4 the following result. Corollary 2.6. If it is consistent that there is an inaccessible cardinal, then it is consistent that ω1 ω1 ω2 ∈ / C(ω1 + 1) but κκ κ+ ∈ C(κ + 1) for every cardinal κ > ω1 . Also, it was shown in [13] that for every specific infinite cardinals κ, λ with κ < λ and κ regular and every specific ordinal δ with κ+ ≤ δ < κ++ and cf(δ) = κ+ , it is consistent that κδ λ ∈ C(δ + 1). Then, the construction carried out in Theorem 2.1 permits us to extend this result to ordinals δ whose cofinality is smaller than κ+ , and hence we obtain the following theorem. Theorem 2.7. Assume that GCH holds. If κ, λ are infinite cardinals with κ < λ and κ regular and δ < κ++ is an ordinal, then in some cardinal-preserving generic extension we have that κδ λ ∈ C(δ + 1). Proof. If δ < κ++ with cf(δ) = κ+ , we have that κδ λ ∈ C(δ + 1) by [13, Theorem 1.2]. So, without loss of generality, we may assume that κ+ < δ < κ++ with cf(δ) ≤ κ. By the main result of [11], there is a κ-closed and κ+ -c.c. partial order P of size κ+ that forces the existence of an LCS space of height δ and width κ. Note that P preserves GCH (see [10, Lemma 5.12 and Theorem 6.14]). Let G be P-generic and let N = V [G]. Suppose that δ is a limit ordinal. Since N satisfies GCH, there is a forcing extension H of N that preserves cofinalities such that in H, 2θ = θ+ for every cardinal θ < cf(δ) and 2cf(δ) ≥ λ (see [10, Theorems 6.14, 6.16 and 6.17]). Now, in H we consider the tree T = 2≤cf(δ) where for s, t ∈ T , s t iff s ⊆ t. Let δα : α < cf(δ) be an increasing sequence of ordinals cofinal in δ. For every non-limit ordinal α < cf(δ) let Zα be a compact scattered space with ht(Zα ) = δα + 1, |Iβ (Zα )| = κ for every ordinal β < δα and |Iδα (Zα )| = 1. Then, as cf(δ) ≤ κ we can proceed as in the proof of part (a) of Theorem 2.1 and show that there is an LCS space X with CS(X) = κδ 2cf(δ) . Now, as λ ≤ 2cf(δ) , we can take an open subspace Z of X such that ht(Z) = ht(X), Iα (Z) = Iα (X) for every α < δ and |Iδ (Z)| = λ. Clearly, Z is locally compact and scattered. So, we have that κδ λ ∈ C(δ + 1). And if δ is a successor ordinal, we consider a cardinal-preserving forcing extension H of N such that (2ω ≥ λ)H , and then we use the argument given in the proof of part (b) of Theorem 2.1. 2
J.C. Martínez / Topology and its Applications 203 (2016) 91–97
95
3. A construction for singular cardinals In this section, we shall show the following result. Theorem 3.1. If κ is a singular cardinal of cofinality ω, then for every cardinal λ such that κ < λ ≤ κω we have that κκ λω2 ∈ C(κ + ω2 ). The following notion will be used in the proof of Theorem 3.1. Definition 3.2. An LCS space X is special, if ht(X) = ht− (X) and for every point x ∈ X there is a compact open neighbourhood U (x) of x such that the following hold: (1) For every x, y ∈ X, U (x) ⊆ U (y) or U (y) ⊆ U (x) or U (x) ∩ U (y) = ∅. (2) If x, y ∈ Iα (X) for some α < ht(X) with x = y, then U (x) ∩ U (y) = ∅. (3) If x ∈ Iα (X) and α < β < ht(X), then there is a y ∈ Iβ (X) such that U (x) ⊆ U (y). By using the argument given in the proof of Proposition 2.5, we can show the following lemma. Lemma 3.3. For every infinite cardinal κ and every non-zero ordinal α < κ+ , there is a special LCS space X such that CS(X) = κα . Proof of Theorem 3.1. Assume that κ is a singular cardinal of cofinality ω. Let κn : n < ω be a strictly increasing sequence of infinite cardinals cofinal in κ. Let λ be a cardinal with κ < λ ≤ κω . We construct an LCS space X of height κ + ω2 such that |Iα (X)| = κ for α < κ and |Iα (X)| = λ for κ ≤ α < κ + ω2 . By Lemma 3.3, we can consider a special LCS space Z0 such that CS(Z0 ) = κκ . For 0 < n < ω we take a homeomorphic copy Zn of Z0 such that Zm ∩ Zn = ∅ for m < n < ω and Zn ∩ ({α} × λ) = ∅ for n < ω and κ ≤ α < κ + ω2 . Then, the underlying set of the required space X will be a set of the form Z ∪ {{α} × λ : κ ≤ α < κ + ω2 } where Z ⊆ {Zn : n ∈ ω}. We will have that Z = {Iα (X) : α < κ} and Iα (X) = {α} × λ for κ ≤ α < κ + ω2 . Fix n < ω. Since Zn is special, for every ordinal α < κ we can pick a point zn,α ∈ Iα (Zn ) in such a way that U (zn,α ) ∩ {U (zn,β ) : β < α} = ∅. Also as λ ≤ κω , by means of a standard combinatorial argument, we can construct a sequence gξ : ξ < λ of functions in ω κ such that gξ (n) > κn for ξ < λ and n < ω and in such a way that the set {n ∈ ω : gμ (n) = gξ (n)} is finite for μ < ξ < λ. Now, in order to define the required space X, first we fix an enumeration without repetitions {yξ : ξ < λ} of {{α} × λ : κ ≤ α < κ + ω2 }. Proceeding by transfinite induction on ξ < λ we can construct an LCS space Yξ such that the following conditions hold: (1) (2) (3) (4)
yξ ∈ Yξ . ht(Yξ ) = o.t.(δ \ κ) + 2 where δ is the ordinal such that yξ ∈ {δ} × λ. For every ordinal ζ < ht(Yξ ), Iζ (Yξ ) is a countable subset of {κ + ζ} × λ. If μ < ξ < λ, then Yμ ∩ Yξ = ∅.
Clearly, it follows from condition (3) that Yξ ∩ {Zn : n ∈ ω} = ∅. Note that the construction of Yξ can be carried out, because for every ordinal α < ω2 we have that ωα ∈ C(α) (see [6]).
J.C. Martínez / Topology and its Applications 203 (2016) 91–97
96
Now, for every ξ < λ we define a space Xξ from the space Yξ and the function gξ as follows. First, we put Wn,ξ = U (zn,gξ (n) ) for every n ∈ ω. And we put Wξ = {Wn,ξ : n ∈ ω}. Let {vk : k ∈ ω} be an enumeration of I0 (Yξ ). Consider a partition {an : n ∈ ω} of ω into infinite subsets. The underlying set of Xξ is Wξ ∪ Yξ . If x ∈ Wn,ξ for some n ∈ ω, then a basic neighbourhood of x in Xξ is an open neighbourhood of x in Wn,ξ (with the relative topology of Zn ). For k ∈ ω, a basic neighbourhood of vk in Xξ is a set of the form {vk } ∪ {Wn,ξ : n ∈ ak , n > m} where m ∈ ω. And if x ∈ Yξ \ I0 (Yξ ), then a basic neighbourhood of x in Xξ is a set of the form U ∪ {Wn,ξ : n ∈ ak , vk ∈ U } where U is an open neighbourhood of x in Yξ . Then, we define the space X as follows. Its underlying set is {Xξ : ξ < λ}, and its topology has {U ⊆ X : U is an open subset of Xξ for some ξ < λ} as a base. It is easy to check that X is locally compact and scattered. To verify that X is Hausdorff, assume that x ∈ Xμ and y ∈ Xξ with x = y and μ = ξ. Without loss of generality, we may assume that x ∈ Yμ and y ∈ Yξ . Let k ∈ ω be such that {n ∈ ω : gμ (n) = gξ (n)} ⊆ k. Let Vx = (Ux ∪ Wμ ) \ (U (z0,gμ (0) ) ∪ . . . ∪ U (zk−1,gμ (k−1) )) where Ux is an open neighbourhood of x in Yμ , and let Vy = (Uy ∪ Wξ ) \ (U (z0,gξ (0) ) ∪ . . . ∪ U (zk−1,gξ (k−1) )) where Uy is an open neighbourhood of y in Yξ . Note that by the way we have chosen the points zn,α , we have Vx ∩ Vy = ∅. Also, it is clear that CS(X) = κκ λω2 . Thus, X is as required. 2 On the other hand, by the main result of [12], we know that it is relatively consistent with ZFC that ωα ∈ C(α) for every ordinal α < ω3 . So, by using an argument parallel to the one given in the proof of Theorem 3.1, we obtain the following consistency result (without using large cardinals). Theorem 3.4. It is relatively consistent with ZFC that for every singular cardinal κ of cofinality ω and every cardinal λ such that κ < λ ≤ κω , κκ λω3 ∈ C(κ + ω3 ). In particular, we obtain the following result from Theorem 3.1 and Theorem 3.4. Corollary 3.5. (a) ℵω ℵω ℵ+ ω ω2 ∈ C(ℵω + ω2 ). (b) It is relatively consistent with ZFC that ℵω ℵω ℵ+ ω ω3 ∈ C(ℵω + ω3 ). Acknowledgements I would like to express my gratitude to Lajos Soukup and the anonymous referee for several useful comments and suggestions. References [1] J. Bagaria, Thin-tall spaces and cardinal sequences, in: E. Pearl (Ed.), Open Problems in Topology II, Elsevier, Amsterdam, 2007, pp. 115–124. [2] J.E. Baumgartner, S. Shelah, Remarks on superatomic Boolean algebras, Ann. Pure Appl. Logic 33 (2) (1987) 109–129. [3] T. Jech, Set Theory, the third millennium edition, Springer Monogr. Math., Springer-Verlag, Berlin, 2003. [4] I. Juhász, S. Shelah, L. Soukup, Z. Szentmiklóssy, A tall space with a small bottom, Proc. Am. Math. Soc. 131 (6) (2003) 1907–1916. [5] I. Juhász, L. Soukup, W. Weiss, Cardinal sequences of length < ω2 under GCH, Fundam. Math. 189 (1) (2006) 35–52. [6] I. Juhász, W. Weiss, On thin-tall scattered spaces, Colloq. Math. 40 (1) (1978/79) 63–68. [7] I. Juhász, W. Weiss, Cardinal sequences, Ann. Pure Appl. Logic 144 (1–3) (2006) 96–106. [8] W. Just, Two consistency results concerning thin-tall Boolean algebras, Algebra Univers. 20 (2) (1985) 135–142. [9] P. Koepke, J.C. Martínez, Superatomic Boolean algebras constructed from morasses, J. Symb. Log. 60 (3) (1995) 940–951. [10] K. Kunen, Set Theory, North-Holland, Amsterdam, 1980.
J.C. Martínez / Topology and its Applications 203 (2016) 91–97
97
[11] J.C. Martínez, A forcing construction of thin-tall Boolean algebras, Fundam. Math. 159 (2) (1999) 99–113. [12] J.C. Martínez, A consistency result on thin-very tall Boolean algebras, Isr. J. Math. 123 (2001) 273–284. [13] J.C. Martínez, L. Soukup, Superatomic Boolean algebras constructed from strongly unbounded functions, Math. Log. Q. 57 (5) (2011) 456–469. [14] J. Roitman, Superatomic Boolean algebras, in: J.D. Monk, R. Bonnet (Eds.), Handbook of Boolean Algebras, vol. 3, North-Holland, Amsterdam, 1989, pp. 719–740.