ORDER-THEORETIC INVARIANTS IN SET-THEORETIC TOPOLOGY

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[10] M. G. Bell, On the combinatorial principle P (c), Fund. Math. 114 (1981), no. 2, 149–157. [11] A. Blass, Combinatorial Cardinal Characteristics of the Continuum, Handbook of Set Theory (M. Foreman, A. Kanamori, and M. Magidor, eds.), Kluwer, to appear. [12] J. Brendle and S. Shelah, Ultrafilters on ω—their ideals and their cardinal characteristics, Trans. AMS 351 (1999), 2643–2674. [13] C. Costantini, A. Fedeli, A. Le Donne, Filters and pathwise connectifications, Rend. Istit. Mat. Univ. Trieste, 32 (2000), no. 1-2, 173–187. [14] Devlin, K., Steprāns, J., and Watson, W. S., The number of directed sets, Rend. Circ. Mat. Palermo (2) 1984, Suppl. No. 4, 31–41. [15] E. van Douwen, Nonhomogeneity of products of preimages and π-weight, Proc. AMS, 69:1 (1978), 183–192. [16] A. Dow and J. Zhou, Two real ultrafilters on ω, Topology Appl. 97 (1999), no. 1-2, 149–154. [17] I. Druzhinina, R. G. Wilson, Pathwise connectifications of OPC spaces, Questions Answers Gen. Topology 20 (2002), no. 1, 75–84. [18] B. A. Efimov, Mappings and imbeddings of dyadic spaces, Mat. Sb. (N.S.) 103 (145) (1977), no. 1, 52–68, 143. [19] A. Emeryk, W. Kulpa, The Sorgenfrey line has no connected compactification, Com- ment. Math. Univ. Carolinae 18 (1977), no. 3, 483–487. 161
[20] R. Engelking, General Topology, Heldermann-Verlag, Berlin, 2nd ed., 1989. [21] A. Fedeli, A. Le Donne, Connectifications and open components, Questions Answers Gen. Topology 18 (2000), no. 1, 41–45. [22] A. Fedeli, A. Le Donne, Dense embeddings in pathwise connected spaces, Topology Appl. 96 (1999), no. 1, 15–22. [23] S. Fuchino, S. Koppelberg, and S. Shelah, Partial orderings with the weak Freese- Nation property, Ann. Pure Appl. Logic 80 (1996), no. 1, 35–54. [24] J. Gerlits, On subspaces of dyadic compacta, Studia Sci. Math. Hungar. 11 (1976), no. 1-2, 115–120. [25] M. Groszek and T. Jech, Generalized iteration of forcing, Trans. Amer. Math. Soc. 324 (1991), no. 1, 1–26. [26] G. Gruenhage, J. Kulesza, A. Le Donne, Connectifications of metrizable space, Topology Appl. 82 (1998), no. 1-3, 171–179. [27] K. P. Hart and G. J. Ridderbos, A note on an example by van Mill, Topology Appl. 150 (2005), no. 1-3, 207-211. [28] J. E. Hart and K. Kunen, Bohr compactifications of discrete structures, Fund. Math. 160 (1999), no. 2, 101–151. [29] F. Hausdorff, Über zei Sätze von G. Fichtenholz und L. Kantorovitch, Studia Math. 6 (1936), 18–19. 162
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ORDER-THEORETIC INVARIANTS IN SET-T
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compactum without reducing its cell
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Contents Abstract i Acknowledgement
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Bibliography 160 vi
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of certain products of homogeneous
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An ordinal α is regular if α = cf
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• regular if for all closed C ⊆
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• locally κ-compact if for every
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The category of boolean algebras is
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B is a base of X if and only if, fo
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A regular cardinal κ is a caliber
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the supremum of the local Noetheria
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• An subset S of M n is definable
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1.7 Forcing Definition 1.7.1. The C
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• M[G] |= ϕ(σ (0) G , . . . ,
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• a reduction of a map g : Q →
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Definition 1.7.16. The product P ×
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If S is a stationary subset of a re
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Chapter 2 Amalgams 2.1 Introduction
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properties to this particular conne
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Theorem 2.2.2. The classes listed b
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Theorem 2.2.3. Suppose X and YS are
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Definition 2.2.7. Suppose W is a su
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y Theorem 2.3.1, ˜ Y is path-conne
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compactum with cellularity c. Moreo
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y(Si) for all i < m. Let W be an op
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Chapter 3 Noetherian types of homog
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Finally, in Section 3.5, we prove s
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Given Theorem 3.2.2, justifying Obs
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and V ∈ Bm and U V . Then m = n
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Definition 3.2.12. Given a space X,
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Proof. We will only prove the first
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Proof. Let A be a neighborhood base
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We may assume there exists an n <
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intervals i∈I (ai, bi) such that
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p ≥ q iff, for all σ ∈ ζ and
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V ∈ V. By Theorem 3.3.2, Q is alm
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Proposition 3.3.11. Suppose a point
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than κ. Then a κ-approximation se
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κ. Let Υ(δγ) = 〈β0, . . . ,
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subset of Gα. For all I ⊆ P(2 κ
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such that V 1 ⊆ W . Therefore, h
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we have A, N , Mα ∈ Mα+1 ≺ H
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We prove this claim by induction. F
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Proof. Let 〈Xi〉i∈I be a seque
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Set A = Aκ∪Aλ∪Ap. Let us show
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exists τ ∈ [Aα]
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Theorem 3.4.3. If κ ≥ ω and B h
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The following theorem is implicit i
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W ⊆ f −1 {0}. By (3) for stage
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3.5 More on local Noetherian type I
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has an open neighborhood Wx that is
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for all σ ∈ [U]
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Lemma 3.5.14. Suppose X is a space,
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πNt(X) ≥ ω1? Question 3.6.3. Su
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Theorem 4.1.3. If X is a homogeneou
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ase of X? Is there such a metric sp
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Set Uα = i
Page 117 and 118: Corollary 4.2.15. Let X be a compac
Page 119 and 120: Corollary 4.2.19 (GCH). There do no
Page 121 and 122: finite cover of X and pairwise ⊆-
Page 123 and 124: w = min([p1, max X] \ U). Then w
Page 125 and 126: points p of Y , there exists α <
Page 127 and 128: x ∈ [ω] ω splits every y ∈ A.
Page 129 and 130: Lemma 5.2.6. Let X be a compact spa
Page 131 and 132: κ ss2 ≤ κ. Every nontrivial fin
Page 133 and 134: Proof. Let P be the κ-long finite
Page 135 and 136: Remark 5.4.2. If P is a finite supp
Page 137 and 138: Let J be the P-name {〈ˇα, pα
Page 139 and 140: are no such α and β, then let Qσ
Page 141 and 142: For each α < λ, let xα be the Co
Page 143 and 144: Proof. See Exercise A12 on page 289
Page 145 and 146: Let Fλα+β to be a Pλα+β-name
Page 147 and 148: Proof. Fix p ∈ ω ∗ . By a resu
Page 149 and 150: χ(〈p〉i
Page 151 and 152: Then χ(p, X) < w(X) or Nt(X) > µ.
Page 153 and 154: Question 5.7.10. Is cf c < χNt(ω
Page 155 and 156: • P ≤T P × Q. • P ≤T R ≥
Page 157 and 158: changed. Moreover, since ω × ω1
Page 159 and 160: Theorem 6.3.7. Assume ♦(E c ω) a
Page 161 and 162: assumption that c = κ + for some c
Page 163 and 164: show that, for a fixed regular unco
Page 165 and 166: and i ∈ dom Γ such that for all
Page 167: Bibliography [1] O. T. Alas, M. G.
Page 171 and 172: [39] K. Kunen, Weak P-points in N
Page 173 and 174: [60] B. É. ˇ Sapirovskiĭ, Cardin
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