ORDER-THEORETIC INVARIANTS IN SET-THEORETIC TOPOLOGY

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ordered lexicographically, has π-weight ω and weight c, and is known to be compact and homogeneous. By Proposition 3.2.22, it has Noetherian type c + . Theorem 3.2.24. Suppose X is a Suslin line. Then πNt(X) ≥ ω1. Proof. Let A be a π-base of X consisting only of open intervals. By Lemma 3.2.20, it suffices to show that A is not ω op -like. Construct a sequence 〈Bn〉n
We may assume there exists an n < ω such that |aα \ ρ| = n for all α < ω1. For each α < ω1, choose a bijection 〈aα,i〉i
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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 ⊆
Page 15 and 16: • locally κ-compact if for every
Page 17 and 18: The category of boolean algebras is
Page 19 and 20: B is a base of X if and only if, fo
Page 21 and 22: A regular cardinal κ is a caliber
Page 23 and 24: the supremum of the local Noetheria
Page 25 and 26: • An subset S of M n is definable
Page 27 and 28: 1.7 Forcing Definition 1.7.1. The C
Page 29 and 30: • M[G] |= ϕ(σ (0) G , . . . ,
Page 31 and 32: • a reduction of a map g : Q →
Page 33 and 34: Definition 1.7.16. The product P ×
Page 35 and 36: If S is a stationary subset of a re
Page 37 and 38: Chapter 2 Amalgams 2.1 Introduction
Page 39 and 40: properties to this particular conne
Page 41 and 42: Theorem 2.2.2. The classes listed b
Page 43 and 44: Theorem 2.2.3. Suppose X and YS are
Page 45 and 46: Definition 2.2.7. Suppose W is a su
Page 47 and 48: y Theorem 2.3.1, ˜ Y is path-conne
Page 49 and 50: compactum with cellularity c. Moreo
Page 51 and 52: y(Si) for all i < m. Let W be an op
Page 53 and 54: Chapter 3 Noetherian types of homog
Page 55 and 56: Finally, in Section 3.5, we prove s
Page 57 and 58: Given Theorem 3.2.2, justifying Obs
Page 59 and 60: and V ∈ Bm and U V . Then m = n
Page 61 and 62: Definition 3.2.12. Given a space X,
Page 63 and 64: Proof. We will only prove the first
Page 65: Proof. Let A be a neighborhood base
Page 69 and 70: intervals i∈I (ai, bi) such that
Page 71 and 72: p ≥ q iff, for all σ ∈ ζ and
Page 73 and 74: V ∈ V. By Theorem 3.3.2, Q is alm
Page 75 and 76: 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
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Corollary 4.2.15. Let X be a compac
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Corollary 4.2.19 (GCH). There do no
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finite cover of X and pairwise ⊆-
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w = min([p1, max X] \ U). Then w
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points p of Y , there exists α <
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x ∈ [ω] ω splits every y ∈ A.
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Lemma 5.2.6. Let X be a compact spa
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κ ss2 ≤ κ. Every nontrivial fin
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Proof. Let P be the κ-long finite
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Remark 5.4.2. If P is a finite supp
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Let J be the P-name {〈ˇα, pα
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are no such α and β, then let Qσ
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For each α < λ, let xα be the Co
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Proof. See Exercise A12 on page 289
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Let Fλα+β to be a Pλα+β-name
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Proof. Fix p ∈ ω ∗ . By a resu
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χ(〈p〉i
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Then χ(p, X) < w(X) or Nt(X) > µ.
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Question 5.7.10. Is cf c < χNt(ω
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• P ≤T P × Q. • P ≤T R ≥
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changed. Moreover, since ω × ω1
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Theorem 6.3.7. Assume ♦(E c ω) a
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assumption that c = κ + for some c
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show that, for a fixed regular unco
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and i ∈ dom Γ such that for all
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Bibliography [1] O. T. Alas, M. G.
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[20] R. Engelking, General Topology
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[39] K. Kunen, Weak P-points in N
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[60] B. É. ˇ Sapirovskiĭ, Cardin
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ORDER-THEORETIC INVARIANTS IN SET-THEORETIC TOPOLOGY

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