ORDER-THEORETIC INVARIANTS IN SET-THEORETIC TOPOLOGY

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Since ψ is easily verified to be a topological embedding, χ(x, ω ∗ ) ≤ χ(ψ(x), ω ∗ ) for all x ∈ ω ∗ . By a result of Pospiˇsil [56], there exists q ∈ ω ∗ such that χ(q, ω ∗ ) = c. Hence, πχ(ψ(q), ω ∗ ) = ω1 and χ(ψ(q), ω ∗ ) = c. By Proposition 3.3.11, Nt(ω ∗ ) > χ(ψ(q), ω ∗ ) = c. Definition 5.2.3. Given n < ω, let ssn (ssω) denote the least cardinal κ for which there exists a sequence 〈fα〉α
Lemma 5.2.6. Let X be a compact space without isolated points and let ω ≤ κ ≤ λ ≤ minp∈X χ(p, X). If X has no 〈λ, κ〉-splitter, then Nt(X) > κ. Proof. Let A be a base of X. Construct a sequence 〈Fα〉α
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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
Page 77 and 78: than κ. Then a κ-approximation se
Page 79 and 80: κ. Let Υ(δγ) = 〈β0, . . . ,
Page 81 and 82: subset of Gα. For all I ⊆ P(2 κ
Page 83 and 84: such that V 1 ⊆ W . Therefore, h
Page 85 and 86: we have A, N , Mα ∈ Mα+1 ≺ H
Page 87 and 88: We prove this claim by induction. F
Page 89 and 90: Proof. Let 〈Xi〉i∈I be a seque
Page 91 and 92: Set A = Aκ∪Aλ∪Ap. Let us show
Page 93 and 94: exists τ ∈ [Aα]
Page 95 and 96: Theorem 3.4.3. If κ ≥ ω and B h
Page 97 and 98: The following theorem is implicit i
Page 99 and 100: W ⊆ f −1 {0}. By (3) for stage
Page 101 and 102: 3.5 More on local Noetherian type I
Page 103 and 104: has an open neighborhood Wx that is
Page 105 and 106: for all σ ∈ [U]
Page 107 and 108: Lemma 3.5.14. Suppose X is a space,
Page 109 and 110: πNt(X) ≥ ω1? Question 3.6.3. Su
Page 111 and 112: Theorem 4.1.3. If X is a homogeneou
Page 113 and 114: ase of X? Is there such a metric sp
Page 115 and 116: 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: x ∈ [ω] ω splits every y ∈ A.
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 and 168: Bibliography [1] O. T. Alas, M. G.
Page 169 and 170: [20] R. Engelking, General Topology
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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ORDER-THEORETIC INVARIANTS IN SET-THEORETIC TOPOLOGY

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