ORDER-THEORETIC INVARIANTS IN SET-THEORETIC TOPOLOGY

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The resolution of concern to us in constructed by van Mill [46]. It is a compactum with weight c, π-weight ω, and character ω1. Moreover, assuming MA + ¬CH (or just p > ω1), this space is homogeneous. (It is not homogeneous if 2 ω < 2 ω1 .) Clearly, this space has sufficiently small Noetherian type and π-type. We just need to show that it has local Noetherian type ω. Van Mill’s space is a resolution of 2 ω at each point into T ω1 where T is the circle group R/Z. Notice that T is metrizable. The following lemma proves that every metric compactum has Noetherian type ω, along with some results that will be useful in Section 3.3. Lemma 3.2.8. Let X be a metric compactum with base A. Then there exists B ⊆ A satisfying the following. 1. B is a base of X. 2. B is ω op -like. 3. If U, V ∈ B and U V , then U ⊆ V . 4. For all Γ ∈ [B]
and V ∈ Bm and U V . Then m = n by minimality of Bn. Also, 0 < diam V because ∅ = U V . Hence, if m > n, then diam V < diam U, in contradiction with U V . Hence, m < n; hence, U ⊆ V . For (1), let p ∈ X and n < ω, and let V be the open ball with radius 2 −n and center p. Then we just need to show that there exists U ∈ B such that p ∈ U ⊆ V . Hence, we may assume {p} ∈ B. Hence, p ∈ En+1; hence, there exists U ∈ Bn+1 such that p ∈ U. Since diam U ≤ 2 −n−1 , we have U ⊆ V . For (2), let n < ω and U ∈ Bn. If U is a singleton, then every superset of U in B is in m≤n Bm. If U is not a singleton, then U has diamater at least 2 −m for some m < ω; whence, every superset of U in B is in l≤m Bl. For (4), suppose Γ ∈ [B]
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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
Page 7 and 8: Bibliography 160 vi
Page 9 and 10: of certain products of homogeneous
Page 11 and 12: An ordinal α is regular if α = cf
Page 13 and 14: • 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: Given Theorem 3.2.2, justifying Obs
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 and 66: Proof. Let A be a neighborhood base
Page 67 and 68: We may assume there exists an n <
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,
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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
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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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