ORDER-THEORETIC INVARIANTS IN SET-THEORETIC TOPOLOGY

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Proof. Let y0, y1 ∈ Y . It suffices to show y1 is in the closure of the component of y0. Let U be a basic open neighborhood of y1. Then there exist n < ω and 〈Si〉i
compactum with cellularity c. Moreover, more careful analysis of the latter space’s connectedness properties shows that it is exceptional. All compact groups are dyadic (Kuz ′ minov [41]), and most other known examples of homogeneous compacta are products of first countable compacta (see Kunen [37] and van Mill [46]). Besides the exceptional homogeneous compactum we shall construct, there is, to the best of the author’s knowledge, only one known construction of an exceptional homogeneous compactum, and its soundness is independent of ZFC. In [46], van Mill constructed a compactum K satisfying π(K) = ω (where π(·) here denotes π-weight) and χ(K) = ω1. Clearly, χ(Z) = ω ≤ π(Z) for all first countable spaces Z. Moreover, Efimov [18] and Gerlits [24] independently proved that πχ(Z) = w(Z) for all dyadic compacta Z. Hence, χ(Z) ≤ π(Z) for all Z homeomorphic to products of dyadic compacta and first countable compacta; hence, K is not homeomorphic to such a product. Under the assumption p > ω1 (which follows from MA+¬CH), van Mill proved that K is homogeneous. However, van Mill also noted that all homogeneous compacta Z satisfy 2 χ(Z) ≤ 2 π(Z) as a corollary of a result of Van Douwen [15]. In particular, if 2 ω < 2 ω1 , then K is not homogeneous. Remark 2.4.2. Hart and Ridderbos’ [27] generalization of van Mill’s construction pro- duces only compacta that have the properties of K listed above. However, van Mill’s K is infinite dimensional, while Hart and Ridderbos produce a zero-dimensional example. It is not clear whether there is a consistently homogeneous compactum Z satisfying 0 < ind Z < ω and π(Z) < χ(Z). Our machine for producing path-connected homogeneous compacta will get us 0 < ind Z < ω, but it will also be easy to see that it entails πχ(Z) ≥ c. Definition 2.4.3. Given a group G acting on a set A with element a, let the stabilizer 42
Page 1 and 2: ORDER-THEORETIC INVARIANTS IN SET-T
Page 3 and 4: compactum without reducing its cell
Page 5 and 6: 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: y Theorem 2.3.1, ˜ Y is path-conne
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 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,
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 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 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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