A Provisional Solution of Nyikos’ Manifold Problem: Hereditarily Normal Manifolds of Dimension > 1 May All Be Metrizable

A Provisional Solution of Nyikos’ Manifold Problem:

Hereditarily Normal Manifolds of Dimension > 1 May All Be Metrizable

(Verification in progress: proofs by Todorcevic and by Dow need to be checked)

Franklin D. Tall

July 23, 2013

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History

I Wilder’s problem

I Rudin’s solution

I Nyikos’ problem

Theorem 1 (Provisional)

If it is consistent there is a supercompact cardinal, it is consistent that every hereditarily normal manifold of dimension>1 is metrizable.

The model

1. Start with a supercompact. Force♦ or add a Cohen real. 2. Construct a coherent Souslin tree S.

3. Iterate proper posets as in proof of consistency of PFA, but only those that preserve S.

4. Force withS.

We sayPFA(S)[S]impliesΦ to mean that if Φ is a proposition, S is a coherent Souslin tree, then any model formed via (3) and (4) is a model of Φ.

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Theorem 1 (Provisional)

If it is consistent there is a supercompact cardinal, it is consistent that every hereditarily normal manifold of dimension>1 is metrizable.

The model

1. Start with a supercompact. Force♦or add a Cohen real.

2. Construct a coherent Souslin tree S.

3. Iterate proper posets as in proof of consistency of PFA, but only those that preserve S.

4. Force withS.

We sayPFA(S)[S]impliesΦ to mean that if Φ is a proposition, S is a coherent Souslin tree, then any model formed via (3) and (4) is a model of Φ.

Theorem 1 (Provisional)

If it is consistent there is a supercompact cardinal, it is consistent that every hereditarily normal manifold of dimension>1 is metrizable.

The model

1. Start with a supercompact. Force♦or add a Cohen real.

2. Construct a coherent Souslin tree S.

3. Iterate proper posets as in proof of consistency of PFA, but only those that preserve S.

4. Force withS.

We sayPFA(S)[S]impliesΦ to mean that if Φ is a proposition, S is a coherent Souslin tree, then any model formed via (3) and (4) is a model of Φ.

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Theorem 1 (Provisional)

If it is consistent there is a supercompact cardinal, it is consistent that every hereditarily normal manifold of dimension>1 is metrizable.

The model

1. Start with a supercompact. Force♦or add a Cohen real.

2. Construct a coherent Souslin tree S.

3. Iterate proper posets as in proof of consistency of PFA, but only those that preserve S.

4. Force withS.

We sayPFA(S)[S]impliesΦ to mean that if Φ is a proposition, S is a coherent Souslin tree, then any model formed via (3) and (4) is a model of Φ.

Theorem 1 (Provisional)

If it is consistent there is a supercompact cardinal, it is consistent that every hereditarily normal manifold of dimension>1 is metrizable.

The model

1. Start with a supercompact. Force♦or add a Cohen real.

2. Construct a coherent Souslin tree S.

3. Iterate proper posets as in proof of consistency of PFA, but only those that preserve S.

4. Force withS.

We sayPFA(S)[S]impliesΦ to mean that if Φ is a proposition, S is a coherent Souslin tree, then any model formed via (3) and (4) is a model of Φ.

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Theorem 1 (Provisional)

If it is consistent there is a supercompact cardinal, it is consistent that every hereditarily normal manifold of dimension>1 is metrizable.

The model

1. Start with a supercompact. Force♦or add a Cohen real.

2. Construct a coherent Souslin tree S.

3. Iterate proper posets as in proof of consistency of PFA, but only those that preserve S.

4. Force withS.

We sayPFA(S)[S]impliesΦ to mean that if Φ is a proposition, S is a coherent Souslin tree, then any model formed via (3) and (4)

PPP−

: In a compact, countably tight space, locally countable subspaces of sizeℵ₁ are σ-discrete.

CW: Normal, first countable spaces areℵ₁-collectionwise Hausdorff.

PPI: Every first countable perfect pre-image of ω₁ includes a copy ofω1.

M-M: Compact, countably tight spaces are sequential.

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PPP−

: In a compact, countably tight space, locally countable subspaces of sizeℵ₁ are σ-discrete.

CW: Normal, first countable spaces areℵ₁-collectionwise Hausdorff.

PPI: Every first countable perfect pre-image of ω₁ includes a copy ofω1.

M-M: Compact, countably tight spaces are sequential.

PPP−

: In a compact, countably tight space, locally countable subspaces of sizeℵ₁ are σ-discrete.

CW: Normal, first countable spaces areℵ₁-collectionwise Hausdorff.

PPI: Every first countable perfect pre-image ofω1 includes a copy ofω1.

M-M: Compact, countably tight spaces are sequential.

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PPP−

: In a compact, countably tight space, locally countable subspaces of sizeℵ₁ are σ-discrete.

CW: Normal, first countable spaces areℵ₁-collectionwise Hausdorff.

PPI: Every first countable perfect pre-image ofω1 includes a copy ofω1.

M-M: Compact, countably tight spaces are sequential.

Definition

A collectionI of countable subsets of a setX is a P-idealif each subset of a member ofI is in I, finite unions of members ofI are inI, and whenever {I_n:n∈ω} ⊆ I, there is a J∈ I such that I_n−J is finite for alln.

I is ℵ₁-generatedif there is{I_α}_α<ω

1 ⊆ I such that for each I ∈ I, there is an α such thatI ⊆I_α.

P₂₂(ℵ₁): SupposeI is an ℵ₁-generated P-ideal on a stationary subsetS ofω₁. Then either:

1. there is a stationary E ⊆S such that every countable subset of E is in I, or

2. there is a stationary D⊆S such that for every countable subsetD1 ofD,D1∩I is finite, for eachI ∈ I.

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Definition

A collectionI of countable subsets of a setX is a P-idealif each subset of a member ofI is in I, finite unions of members ofI are inI, and whenever {I_n:n∈ω} ⊆ I, there is a J∈ I such that I_n−J is finite for alln.

I is ℵ₁-generatedif there is{I_α}_α<ω

1 ⊆ I such that for each I ∈ I, there is an α such thatI ⊆I_α.

P₂₂(ℵ₁): SupposeI is an ℵ₁-generated P-ideal on a stationary subsetS ofω₁. Then either:

1. there is a stationary E ⊆S such that every countable subset of E is in I, or

2. there is a stationary D⊆S such that for every countable subsetD1 ofD,D1∩I is finite, for eachI ∈ I.

Definition

A collectionI of countable subsets of a setX is a P-idealif each subset of a member ofI is in I, finite unions of members ofI are inI, and whenever {I_n:n∈ω} ⊆ I, there is a J∈ I such that I_n−J is finite for alln.

I is ℵ₁-generatedif there is{I_α}_α<ω

1 ⊆ I such that for each I ∈ I, there is an α such thatI ⊆I_α.

P₂₂(ℵ₁): SupposeI is an ℵ₁-generated P-ideal on a stationary subsetS ofω₁. Then either:

1. there is a stationary E ⊆S such that every countable subset of E is in I, or

2. there is a stationary D⊆S such that for every countable subsetD1 ofD,D1∩I is finite, for eachI ∈ I.

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Definition

A collectionI of countable subsets of a setX is a P-idealif each subset of a member ofI is in I, finite unions of members ofI are inI, and whenever {I_n:n∈ω} ⊆ I, there is a J∈ I such that I_n−J is finite for alln.

I is ℵ₁-generatedif there is{I_α}_α<ω

1 ⊆ I such that for each I ∈ I, there is an α such thatI ⊆I_α.

P₂₂(ℵ₁): SupposeI is an ℵ₁-generated P-ideal on a stationary subsetS ofω₁. Then either:

1. there is a stationary E ⊆S such that every countable subset of E is in I, or

2. there is a stationary D⊆S such that for every countable subsetD1 ofD,D1∩I is finite, for eachI ∈ I.

Definition

A collectionI of countable subsets of a setX is a P-idealif each subset of a member ofI is in I, finite unions of members ofI are inI, and whenever {I_n:n∈ω} ⊆ I, there is a J∈ I such that I_n−J is finite for alln.

I is ℵ₁-generatedif there is{I_α}_α<ω

1 ⊆ I such that for each I ∈ I, there is an α such thatI ⊆I_α.

P₂₂(ℵ₁): SupposeI is an ℵ₁-generated P-ideal on a stationary subsetS ofω₁. Then either:

1. there is a stationary E ⊆S such that every countable subset of E is in I, or

2. there is a stationary D⊆S such that for every countable subsetD1 ofD,D1∩I is finite, for eachI ∈ I.

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Theorem 2 PPP−

+CW+PPI+P₂₂(ℵ₁)implies every T₅ manifold of dimension>1 is metrizable.

Theorem 3 (Provisional) PFA(S)[S]impliesPPP−

,CW,PPI, and P22(ℵ₁). Proof of Theorem 2.

Assemble pieces from Balogh’s and Nyikos’ papers.

Theorem 2 PPP−

+CW+PPI+P₂₂(ℵ₁)implies every T₅ manifold of dimension>1 is metrizable.

Theorem 3 (Provisional) PFA(S)[S] impliesPPP−

,CW,PPI, and P22(ℵ₁).

Proof of Theorem 2.

Assemble pieces from Balogh’s and Nyikos’ papers.

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Proof of Theorem 3.

I PFA(S)[S] implies M-M [Tod]: Claimed by Todorcevic.

I PFA(S)[S] impliesPPP−

, assuming M-M [FTT].

I PFA(S)[S] implies CW [LT10]:

I PFA(S)[S] implies P22(ℵ₁) (indeed P-ideal Dichotomy) [Tod].

I PFA(S)[S] implies PPI [Dow]: Claimed by Dow.

I Steps remaining: Check Dow’s proof; Todorcevic fills gap in his proof that PFA(S)[S] impliesM-M.

Proof of Theorem 3.

I PFA(S)[S] implies M-M [Tod]: Claimed by Todorcevic.

I PFA(S)[S] impliesPPP−

, assuming M-M [FTT].

I PFA(S)[S] implies CW [LT10]:

I PFA(S)[S] implies P22(ℵ₁) (indeed P-ideal Dichotomy) [Tod].

I PFA(S)[S] implies PPI [Dow]: Claimed by Dow.

I Steps remaining: Check Dow’s proof; Todorcevic fills gap in his proof that PFA(S)[S] impliesM-M.

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Proof of Theorem 3.

I PFA(S)[S] implies M-M [Tod]: Claimed by Todorcevic.

I PFA(S)[S] impliesPPP−

, assuming M-M [FTT].

I PFA(S)[S] implies CW [LT10]:

I PFA(S)[S] implies P22(ℵ₁) (indeed P-ideal Dichotomy) [Tod].

I PFA(S)[S] implies PPI [Dow]: Claimed by Dow.

I Steps remaining: Check Dow’s proof; Todorcevic fills gap in his proof that PFA(S)[S] impliesM-M.

Proof of Theorem 3.

I PFA(S)[S] implies M-M [Tod]: Claimed by Todorcevic.

I PFA(S)[S] impliesPPP−

, assuming M-M [FTT].

I PFA(S)[S] implies CW [LT10]:

I PFA(S)[S] implies P22(ℵ₁) (indeed P-ideal Dichotomy) [Tod].

I PFA(S)[S] implies PPI [Dow]: Claimed by Dow.

I Steps remaining: Check Dow’s proof; Todorcevic fills gap in his proof that PFA(S)[S] impliesM-M.

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Proof of Theorem 3.

I PFA(S)[S] implies M-M [Tod]: Claimed by Todorcevic.

I PFA(S)[S] impliesPPP−

, assuming M-M [FTT].

I PFA(S)[S] implies CW [LT10]:

I PFA(S)[S] implies P22(ℵ₁) (indeed P-ideal Dichotomy) [Tod].

I PFA(S)[S] implies PPI [Dow]: Claimed by Dow.

I Steps remaining: Check Dow’s proof; Todorcevic fills gap in his proof that PFA(S)[S] impliesM-M.

Proof of Theorem 3.

I PFA(S)[S] implies M-M [Tod]: Claimed by Todorcevic.

I PFA(S)[S] impliesPPP−

, assuming M-M [FTT].

I PFA(S)[S] implies CW [LT10]:

I PFA(S)[S] implies P22(ℵ₁) (indeed P-ideal Dichotomy) [Tod].

I PFA(S)[S] implies PPI [Dow]: Claimed by Dow.

I Steps remaining: Check Dow’s proof; Todorcevic fills gap in his proof that PFA(S)[S] impliesM-M.

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Aside

PFA(S)[S] also implies:

1. b=ℵ₂ = 2^ℵ0,

2. p=ℵ₁,

3. there are no first countableL-spaces [LT02] , 4. there are no compactS-spaces [Tod],

5. locally compact normal spaces are ℵ₁-collectionwise Hausdorff [Tal],

6. compact spaces withT₅ squares are metrizable [LT02]. By doing a preliminary forcing, one can get a model in which also:

7. normal spaces which are either first countable or locally compact are collectionwise Hausdorff,

8. locally compact perfectly normal spaces are paracompact [LT10].

Aside

PFA(S)[S] also implies:

1. b=ℵ₂ = 2^ℵ0, 2. p=ℵ₁,

3. there are no first countableL-spaces [LT02] , 4. there are no compactS-spaces [Tod],

5. locally compact normal spaces are ℵ₁-collectionwise Hausdorff [Tal],

6. compact spaces withT₅ squares are metrizable [LT02]. By doing a preliminary forcing, one can get a model in which also:

7. normal spaces which are either first countable or locally compact are collectionwise Hausdorff,

8. locally compact perfectly normal spaces are paracompact [LT10].

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Aside

PFA(S)[S] also implies:

1. b=ℵ₂ = 2^ℵ0, 2. p=ℵ₁,

3. there are no first countableL-spaces [LT02]

, 4. there are no compactS-spaces [Tod],

5. locally compact normal spaces are ℵ₁-collectionwise Hausdorff [Tal],

6. compact spaces withT₅ squares are metrizable [LT02]. By doing a preliminary forcing, one can get a model in which also:

7. normal spaces which are either first countable or locally compact are collectionwise Hausdorff,

8. locally compact perfectly normal spaces are paracompact [LT10].

Aside

PFA(S)[S] also implies:

1. b=ℵ₂ = 2^ℵ0, 2. p=ℵ₁,

3. there are no first countableL-spaces [LT02] , 4. there are no compactS-spaces [Tod],

5. locally compact normal spaces are ℵ₁-collectionwise Hausdorff [Tal],

6. compact spaces withT₅ squares are metrizable [LT02]. By doing a preliminary forcing, one can get a model in which also:

7. normal spaces which are either first countable or locally compact are collectionwise Hausdorff,

8. locally compact perfectly normal spaces are paracompact [LT10].

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Aside

PFA(S)[S] also implies:

1. b=ℵ₂ = 2^ℵ0, 2. p=ℵ₁,

3. there are no first countableL-spaces [LT02] , 4. there are no compactS-spaces [Tod],

5. locally compact normal spaces are ℵ₁-collectionwise Hausdorff [Tal],

6. compact spaces withT₅ squares are metrizable [LT02]. By doing a preliminary forcing, one can get a model in which also:

7. normal spaces which are either first countable or locally compact are collectionwise Hausdorff,

8. locally compact perfectly normal spaces are paracompact [LT10].

Aside

PFA(S)[S] also implies:

1. b=ℵ₂ = 2^ℵ0, 2. p=ℵ₁,

3. there are no first countableL-spaces [LT02] , 4. there are no compactS-spaces [Tod],

5. locally compact normal spaces are ℵ₁-collectionwise Hausdorff [Tal],

6. compact spaces withT₅ squares are metrizable [LT02].

By doing a preliminary forcing, one can get a model in which also: 7. normal spaces which are either first countable or locally

compact are collectionwise Hausdorff,

8. locally compact perfectly normal spaces are paracompact [LT10].

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Aside

PFA(S)[S] also implies:

1. b=ℵ₂ = 2^ℵ0, 2. p=ℵ₁,

3. there are no first countableL-spaces [LT02] , 4. there are no compactS-spaces [Tod],

5. locally compact normal spaces are ℵ₁-collectionwise Hausdorff [Tal],

6. compact spaces withT₅ squares are metrizable [LT02].

By doing a preliminary forcing, one can get a model in which also:

7. normal spaces which are either first countable or locally compact are collectionwise Hausdorff,

8. locally compact perfectly normal spaces are paracompact [LT10].

Aside

PFA(S)[S] also implies:

1. b=ℵ₂ = 2^ℵ0, 2. p=ℵ₁,

3. there are no first countableL-spaces [LT02] , 4. there are no compactS-spaces [Tod],

5. locally compact normal spaces are ℵ₁-collectionwise Hausdorff [Tal],

6. compact spaces withT₅ squares are metrizable [LT02].

By doing a preliminary forcing, one can get a model in which also:

7. normal spaces which are either first countable or locally compact are collectionwise Hausdorff,

8. locally compact perfectly normal spaces are paracompact [LT10].

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The topology Definition

A locally compact spaceX is of Type I if it can be expressed as X =S

α<ω1M_α, where each M_α is open, M_α is Lindel¨of and included inMα+1, and for limit α,Mα =S

β<αMβ. {M_α:α < ω₁} is called acanonical sequence for M.

For a manifold, we may assume eachMα is Lindel¨of.

The topology Definition

A locally compact spaceX is of Type I if it can be expressed as X =S

α<ω1M_α, where each M_α is open, M_α is Lindel¨of and included inMα+1, and for limit α,Mα =S

β<αMβ. {M_α:α < ω₁} is called acanonical sequence for M.

For a manifold, we may assume eachMα is Lindel¨of.

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Lemma 2 P

PP−

+CW implies T5 manifolds are of Type I.

Proof.

The manifoldM has a basis of open Lindel¨of subspaces. We will show closures of Lindel¨of subspaces are Lindel¨of. Then we can define open Lindel¨of M_α,α < ω₁, by recursion:

Start withM₀, cover M₀ by open Lindel¨of sets, take countable subcover, take closure of union, etc. By first countable,S

α<ω1Mα

is clopen and so =M.

To showY Lindel¨of implies Y Lindel¨of, noteY is metrizable, hence separable, soY separable. Y isT5, so CWimplies it has countable spread, as does its one-point compactificationY^∗. Y^∗ is countably tight; it is hereditarily Lindel¨of, for byPPP−

, an uncountable right-separated subspace would beσ-discrete, contradiction.

Lemma 2 P

PP−

+CW implies T5 manifolds are of Type I.

Proof.

The manifoldM has a basis of open Lindel¨of subspaces. We will show closures of Lindel¨of subspaces are Lindel¨of.

Then we can define open Lindel¨of M_α,α < ω₁, by recursion:

Start withM₀, cover M₀ by open Lindel¨of sets, take countable subcover, take closure of union, etc. By first countable,S

α<ω1Mα

is clopen and so =M.

To showY Lindel¨of implies Y Lindel¨of, noteY is metrizable, hence separable, soY separable. Y isT5, so CWimplies it has countable spread, as does its one-point compactificationY^∗. Y^∗ is countably tight; it is hereditarily Lindel¨of, for byPPP−

, an uncountable right-separated subspace would beσ-discrete, contradiction.

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Lemma 2 P

PP−

+CW implies T5 manifolds are of Type I.

Proof.

The manifoldM has a basis of open Lindel¨of subspaces. We will show closures of Lindel¨of subspaces are Lindel¨of. Then we can define open Lindel¨of M_α,α < ω₁, by recursion:

Start withM₀, cover M₀ by open Lindel¨of sets, take countable subcover, take closure of union, etc. By first countable,S

α<ω1Mα

is clopen and so =M.

To showY Lindel¨of implies Y Lindel¨of, noteY is metrizable, hence separable, soY separable. Y isT5, so CWimplies it has countable spread, as does its one-point compactificationY^∗. Y^∗ is countably tight; it is hereditarily Lindel¨of, for byPPP−

, an uncountable right-separated subspace would beσ-discrete, contradiction.

Lemma 2 P

PP−

+CW implies T5 manifolds are of Type I.

Proof.

The manifoldM has a basis of open Lindel¨of subspaces. We will show closures of Lindel¨of subspaces are Lindel¨of. Then we can define open Lindel¨of M_α,α < ω₁, by recursion:

Start withM₀, cover M₀ by open Lindel¨of sets, take countable subcover, take closure of union, etc.

By first countable, S

α<ω1Mα

is clopen and so =M.

To showY Lindel¨of implies Y Lindel¨of, noteY is metrizable, hence separable, soY separable. Y isT5, so CWimplies it has countable spread, as does its one-point compactificationY^∗. Y^∗ is countably tight; it is hereditarily Lindel¨of, for byPPP−

, an uncountable right-separated subspace would beσ-discrete, contradiction.

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Lemma 2 P

PP−

+CW implies T5 manifolds are of Type I.

Proof.

The manifoldM has a basis of open Lindel¨of subspaces. We will show closures of Lindel¨of subspaces are Lindel¨of. Then we can define open Lindel¨of M_α,α < ω₁, by recursion:

Start withM₀, cover M₀ by open Lindel¨of sets, take countable subcover, take closure of union, etc. By first countable,S

α<ω1Mα

is clopen and so =M.

To showY Lindel¨of implies Y Lindel¨of, noteY is metrizable, hence separable, soY separable. Y isT5, so CWimplies it has countable spread, as does its one-point compactificationY^∗. Y^∗ is countably tight; it is hereditarily Lindel¨of, for byPPP−

, an uncountable right-separated subspace would beσ-discrete, contradiction.

Lemma 2 P

PP−

+CW implies T5 manifolds are of Type I.

Proof.

The manifoldM has a basis of open Lindel¨of subspaces. We will show closures of Lindel¨of subspaces are Lindel¨of. Then we can define open Lindel¨of M_α,α < ω₁, by recursion:

Start withM₀, cover M₀ by open Lindel¨of sets, take countable subcover, take closure of union, etc. By first countable,S

α<ω1Mα

is clopen and so =M.

To showY Lindel¨of implies Y Lindel¨of, noteY is metrizable, hence separable, soY separable.

Y isT5, so CWimplies it has countable spread, as does its one-point compactificationY^∗. Y^∗ is countably tight; it is hereditarily Lindel¨of, for byPPP−

, an uncountable right-separated subspace would beσ-discrete, contradiction.

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Lemma 2 P

PP−

+CW implies T5 manifolds are of Type I.

Proof.

The manifoldM has a basis of open Lindel¨of subspaces. We will show closures of Lindel¨of subspaces are Lindel¨of. Then we can define open Lindel¨of M_α,α < ω₁, by recursion:

Start withM₀, cover M₀ by open Lindel¨of sets, take countable subcover, take closure of union, etc. By first countable,S

α<ω1Mα

is clopen and so =M.

To showY Lindel¨of implies Y Lindel¨of, noteY is metrizable, hence separable, soY separable. Y isT₅, so CWimplies it has countable spread, as does its one-point compactificationY^∗.

Y^∗ is countably tight; it is hereditarily Lindel¨of, for byPPP−

, an uncountable right-separated subspace would beσ-discrete, contradiction.

Lemma 2 P

PP−

+CW implies T5 manifolds are of Type I.

Proof.

The manifoldM has a basis of open Lindel¨of subspaces. We will show closures of Lindel¨of subspaces are Lindel¨of. Then we can define open Lindel¨of M_α,α < ω₁, by recursion:

Start withM₀, cover M₀ by open Lindel¨of sets, take countable subcover, take closure of union, etc. By first countable,S

α<ω1Mα

is clopen and so =M.

To showY Lindel¨of implies Y Lindel¨of, noteY is metrizable, hence separable, soY separable. Y isT₅, so CWimplies it has countable spread, as does its one-point compactificationY^∗. Y^∗ is countably tight; it is hereditarily Lindel¨of, for byPPP−

, an uncountable right-separated subspace would beσ-discrete, contradiction.

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Lemma 3 PPP−

+CW implies a locally compact subspace of a T₅ manifold is paracompact if and only if it does not include a perfect pre-image ofω₁.

Proof. Postponed.

Lemma 3 PPP−

+CW implies a locally compact subspace of a T₅ manifold is paracompact if and only if it does not include a perfect pre-image ofω₁.

Proof.

Postponed.

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Lemma 3 PPP−

+CW implies a locally compact subspace of a T₅ manifold is paracompact if and only if it does not include a perfect pre-image ofω₁.

Proof.

Postponed.

In fact, if Todorcevic’s M-M result is correct, Larson and I can prove

Theorem ([LTa], Provisional)

If it’s consistent there is a supercompact, it’s consistent that a locally compact T₅ space is (hereditarily) paracompact iff it does not include a perfect pre-image ofω1.

Adding Dow, I can get Theorem

If it’s consistent there is a supercompact, it’s consistent that a locally compact T₅ space is paracompact iff it does not include a copy ofω1.

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In fact, if Todorcevic’s M-M result is correct, Larson and I can prove

Theorem ([LTa], Provisional)

If it’s consistent there is a supercompact, it’s consistent that a locally compact T₅ space is (hereditarily) paracompact iff it does not include a perfect pre-image ofω1.

Adding Dow, I can get Theorem

If it’s consistent there is a supercompact, it’s consistent that a locally compact T₅ space is paracompact iff it does not include a copy ofω1.

Definition

Supposeπ:X →ω1. We sayY ⊆X is unbounded ifπ(Y) is unbounded.

Lemma 6 (Nyi02) PPI+PPP−

+CWimplies a T5, perfect pre-image of ω1 included in a manifold is the union of a paracompact space with a finite number of disjoint unbounded copies ofω1.

Definition

A selection of one point from each non-emptyB_α=M_α−M_α, where{M_α :α < ω1} is a canonical sequence for a Type I space is called abone-scan.

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Definition

Supposeπ:X →ω1. We sayY ⊆X is unbounded ifπ(Y) is unbounded.

Lemma 6 (Nyi02) PPI+PPP−

+CWimplies a T5, perfect pre-image of ω1 included in a manifold is the union of a paracompact space with a finite number of disjoint unbounded copies ofω1.

Definition

A selection of one point from each non-emptyB_α=M_α−M_α, where{M_α :α < ω1} is a canonical sequence for a Type I space is called abone-scan.

Definition

Supposeπ:X →ω1. We sayY ⊆X is unbounded ifπ(Y) is unbounded.

Lemma 6 (Nyi02) PPI+PPP−

+CWimplies a T5, perfect pre-image of ω1 included in a manifold is the union of a paracompact space with a finite number of disjoint unbounded copies ofω1.

Definition

A selection of one point from each non-emptyB_α=M_α−M_α, where{M_α :α < ω1} is a canonical sequence for a Type I space is called abone-scan.

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Lemma 4

Suppose S is a stationary subset ofω₁ and Y ={y_α :α∈S}is a subset of a bone-scan of a canonical sequence for a Type I, normal, first countable space M, such that countable subsets of Y have compact closure in M. Then Y is a perfect pre-image ofω₁.

Proof. Postponed.

Lemma 4

Suppose S is a stationary subset ofω₁ and Y ={y_α :α∈S}is a subset of a bone-scan of a canonical sequence for a Type I, normal, first countable space M, such that countable subsets of Y have compact closure in M. Then Y is a perfect pre-image ofω₁.

Proof.

Postponed.

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Some proofs Lemma 6

PPI+PPP−

+CWimplies a T₅, perfect pre-image of ω₁ included in a manifold is the union of a paracompact space with a finite number of disjoint unbounded copies ofω1.

Proof.

By first countable andPPI,X ⊇W1∼=ω1. ClaimW1 is

unbounded. If not,W1 ⊆π⁻¹([0, α]) for some α. W1 is countably compact in first countableX, so closed in X and hence in

π⁻¹([0, α]), which is compact. But thenW1 is compact, contradiction.

Some proofs Lemma 6

PPI+PPP−

+CWimplies a T₅, perfect pre-image of ω₁ included in a manifold is the union of a paracompact space with a finite number of disjoint unbounded copies ofω1.

Proof.

By first countable andPPI,X ⊇W1∼=ω1. ClaimW1 is unbounded.

If not, W1 ⊆π⁻¹([0, α]) for some α. W1 is countably compact in first countableX, so closed in X and hence in

π⁻¹([0, α]), which is compact. But thenW1 is compact, contradiction.

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Some proofs Lemma 6

PPI+PPP−

+CWimplies a T₅, perfect pre-image of ω₁ included in a manifold is the union of a paracompact space with a finite number of disjoint unbounded copies ofω1.

Proof.

By first countable andPPI,X ⊇W1∼=ω1. ClaimW1 is

unbounded. If not,W1 ⊆π⁻¹([0, α]) for some α. W1 is countably compact in first countableX, so closed in X and hence in

π⁻¹([0, α]), which is compact.

But thenW1 is compact, contradiction.

Some proofs Lemma 6

PPI+PPP−

+CWimplies a T₅, perfect pre-image of ω₁ included in a manifold is the union of a paracompact space with a finite number of disjoint unbounded copies ofω1.

Proof.

By first countable andPPI,X ⊇W1∼=ω1. ClaimW1 is

unbounded. If not,W1 ⊆π⁻¹([0, α]) for some α. W1 is countably compact in first countableX, so closed in X and hence in

π⁻¹([0, α]), which is compact. But then W1 is compact, contradiction.

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SinceW₁ is closed,X −W₁ is open and hence locally compact.

If it is paracompact, we are done; if not, apply Lemma 3 again to get perfect pre-imageP ofω₁ included inX −W₁. By PPI, take a copyW₂ of ω₁ included in P. Continue. We must end at some finite stage, since:

Lemma 7 ( [Nyi04a])

Let X be a T₅ space,π :X →ω₁ continuous, π⁻¹({α}) countably compact for allα∈S , a stationary subset of ω1. Then X cannot include an infinite disjoint family of closed, countably compact, unbounded subspaces.

SinceW₁ is closed,X −W₁ is open and hence locally compact. If it is paracompact, we are done; if not, apply Lemma 3 again to get perfect pre-imageP of ω₁ included inX −W₁.

By PPI, take a copyW₂ of ω₁ included in P. Continue. We must end at some finite stage, since:

Lemma 7 ( [Nyi04a])

Let X be a T₅ space,π :X →ω₁ continuous, π⁻¹({α}) countably compact for allα∈S , a stationary subset of ω1. Then X cannot include an infinite disjoint family of closed, countably compact, unbounded subspaces.

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SinceW₁ is closed,X −W₁ is open and hence locally compact. If it is paracompact, we are done; if not, apply Lemma 3 again to get perfect pre-imageP of ω₁ included inX −W₁. By PPI, take a copyW₂ of ω₁ included in P. Continue. We must end at some finite stage, since:

Lemma 7 ( [Nyi04a])

Let X be a T₅ space,π :X →ω₁ continuous, π⁻¹({α}) countably compact for allα∈S , a stationary subset of ω1. Then X cannot include an infinite disjoint family of closed, countably compact, unbounded subspaces.

SinceW₁ is closed,X −W₁ is open and hence locally compact. If it is paracompact, we are done; if not, apply Lemma 3 again to get perfect pre-imageP of ω₁ included inX −W₁. By PPI, take a copyW₂ of ω₁ included in P. Continue. We must end at some finite stage, since:

Lemma 7 ( [Nyi04a])

Let X be a T₅ space,π :X →ω₁ continuous, π⁻¹({α}) countably compact for allα∈S , a stationary subset of ω1. Then X cannot include an infinite disjoint family of closed, countably compact, unbounded subspaces.

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SinceW₁ is closed,X −W₁ is open and hence locally compact. If it is paracompact, we are done; if not, apply Lemma 3 again to get perfect pre-imageP of ω₁ included inX −W₁. By PPI, take a copyW₂ of ω₁ included in P. Continue. We must end at some finite stage, since:

Lemma 7 ( [Nyi04a])

Let X be a T₅ space,π :X →ω₁ continuous, π⁻¹({α}) countably compact for allα∈S , a stationary subset of ω1. Then X cannot include an infinite disjoint family of closed, countably compact, unbounded subspaces.

The only use of dim>1 is what I call theFat Boundary Theorem:

Lemma 8 ( [Nyi02])

If M is a Type I manifold of dim>1, then there is a canonical sequence{M_α}_α<ω1 for M such that for each p∈B_α =M_α−M_α, there is a non-trivial continuum Kα(p)⊆Bα, with p ∈Kα(p).

17 / 26

The only use of dim>1 is what I call theFat Boundary Theorem:

Lemma 8 ( [Nyi02])

If M is a Type I manifold of dim>1, then there is a canonical sequence{M_α}_α<ω1 for M such that for each p∈B_α =M_α−M_α, there is a non-trivial continuum Kα(p)⊆Bα, with p ∈Kα(p).

The only use of dim>1 is what I call theFat Boundary Theorem:

Lemma 8 ( [Nyi02])

If M is a Type I manifold of dim>1, then there is a canonical sequence{M_α}_α<ω1 for M such that for each p∈B_α =M_α−M_α, there is a non-trivial continuum Kα(p)⊆Bα, with p ∈Kα(p).

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Ideals enter the picture via:

Lemma 11 ( [Nyi02])

Let M be a hereditarilyℵ₁-collectionwise Hausdorff Type I

subspace of a manifold, and{y_α:α∈S}, S a stationary subset of ω1, be a subset of a bone-scan. Then P₂₂(ℵ₁)implies there is a stationary S⁰⊆S such that every countable subset of

{y_α :α∈S⁰} has compact closure in M.

Ideals enter the picture via:

Lemma 11 ( [Nyi02])

Let M be a hereditarilyℵ₁-collectionwise Hausdorff Type I

subspace of a manifold, and{y_α:α∈S}, S a stationary subset of ω1, be a subset of a bone-scan. Then P₂₂(ℵ₁)implies there is a stationary S⁰⊆S such that every countable subset of

{y_α :α∈S⁰} has compact closure in M.

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Ideals enter the picture via:

Lemma 11 ( [Nyi02])

Let M be a hereditarilyℵ₁-collectionwise Hausdorff Type I

subspace of a manifold, and{y_α:α∈S}, S a stationary subset of ω1, be a subset of a bone-scan. Then P₂₂(ℵ₁)implies there is a stationary S⁰⊆S such that every countable subset of

{y_α :α∈S⁰} has compact closure in M.

We also need:

Lemma 12 P

PP−

→ ¬CH.

Proof.

CH→ ∃ compactS-space: the Kunen Line [JKR].PPP−

implies there are no compactS-spaces, since S-spaces are countably tight.

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We also need:

Lemma 12 P

PP−

→ ¬CH.

Proof.

CH→ ∃ compactS-space: the Kunen Line [JKR].PPP−

implies there are no compactS-spaces, since S-spaces are countably tight.

We also need:

Lemma 12 P

PP−

→ ¬CH.

Proof.

CH→ ∃ compactS-space: the Kunen Line [JKR].PPP−

implies there are no compactS-spaces, since S-spaces are countably tight.

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Now we can finally prove:

Theorem 2 P

PP−

+CW+PPI+P22(ℵ₁) implies T5 manifolds of dimension greater than1are metrizable.

Proof.

Non-metrizableT5 Type I manifoldM includes a perfect pre-image ofω₁, and hence a copy W ofω₁. W must meet stationarily many B_α, else it would be a closed subspace of a sum of Lindel¨of spaces and so would be paracompact. Pickwα∈W ∩Bα,α∈some stationaryE₀⊆ω₁. By Lemma 8, pick non-trivial continuaK_α such thatw_α ∈K_α⊆B_α. |W|=ℵ₁ <2^ℵ0, so pickq_α ∈K_α−W. LetM_α⁰ =Mα−W. W is closed, so M−W is open, so locally compact. M_α is hereditarily Lindel¨of, soM−W =S

α<ω1M_α⁰ is Type I.

Now we can finally prove:

Theorem 2 P

PP−

+CW+PPI+P22(ℵ₁)implies T5 manifolds of dimension greater than1are metrizable.

Proof.

Non-metrizableT5 Type I manifoldM includes a perfect pre-image ofω₁, and hence a copy W ofω₁. W must meet stationarily many B_α, else it would be a closed subspace of a sum of Lindel¨of spaces and so would be paracompact. Pickwα∈W ∩Bα,α∈some stationaryE₀⊆ω₁. By Lemma 8, pick non-trivial continuaK_α such thatw_α ∈K_α⊆B_α. |W|=ℵ₁ <2^ℵ0, so pickq_α ∈K_α−W. LetM_α⁰ =Mα−W. W is closed, so M−W is open, so locally compact. M_α is hereditarily Lindel¨of, soM−W =S

α<ω1M_α⁰ is Type I.

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Now we can finally prove:

Theorem 2 P

PP−

+CW+PPI+P22(ℵ₁)implies T5 manifolds of dimension greater than1are metrizable.

Proof.

Non-metrizableT5 Type I manifoldM includes a perfect pre-image ofω₁, and hence a copy W ofω₁.

W must meet stationarily many B_α, else it would be a closed subspace of a sum of Lindel¨of spaces and so would be paracompact. Pickwα∈W ∩Bα,α∈some stationaryE₀⊆ω₁. By Lemma 8, pick non-trivial continuaK_α such thatw_α ∈K_α⊆B_α. |W|=ℵ₁ <2^ℵ0, so pickq_α ∈K_α−W. LetM_α⁰ =Mα−W. W is closed, so M−W is open, so locally compact. M_α is hereditarily Lindel¨of, soM−W =S

α<ω1M_α⁰ is Type I.

Now we can finally prove:

Theorem 2 P

PP−

+CW+PPI+P22(ℵ₁)implies T5 manifolds of dimension greater than1are metrizable.

Proof.

Non-metrizableT5 Type I manifoldM includes a perfect pre-image ofω₁, and hence a copy W ofω₁. W must meet stationarily many Bα, else it would be a closed subspace of a sum of Lindel¨of spaces and so would be paracompact.

Pickwα∈W ∩Bα,α∈some stationaryE₀⊆ω₁. By Lemma 8, pick non-trivial continuaK_α such thatw_α ∈K_α⊆B_α. |W|=ℵ₁ <2^ℵ0, so pickq_α ∈K_α−W. LetM_α⁰ =Mα−W. W is closed, so M−W is open, so locally compact. M_α is hereditarily Lindel¨of, soM−W =S

α<ω1M_α⁰ is Type I.

20 / 26

Now we can finally prove:

Theorem 2 P

PP−

+CW+PPI+P22(ℵ₁)implies T5 manifolds of dimension greater than1are metrizable.

Proof.

Non-metrizableT5 Type I manifoldM includes a perfect pre-image ofω₁, and hence a copy W ofω₁. W must meet stationarily many Bα, else it would be a closed subspace of a sum of Lindel¨of spaces and so would be paracompact. Pickwα∈W ∩Bα,α∈some stationaryE₀⊆ω₁.

By Lemma 8, pick non-trivial continuaK_α such thatw_α ∈K_α⊆B_α. |W|=ℵ₁ <2^ℵ0, so pickq_α ∈K_α−W. LetM_α⁰ =Mα−W. W is closed, so M−W is open, so locally compact. M_α is hereditarily Lindel¨of, soM−W =S

α<ω1M_α⁰ is Type I.

Now we can finally prove:

Theorem 2 P

PP−

+CW+PPI+P22(ℵ₁)implies T5 manifolds of dimension greater than1are metrizable.

Proof.

Non-metrizableT5 Type I manifoldM includes a perfect pre-image ofω₁, and hence a copy W ofω₁. W must meet stationarily many Bα, else it would be a closed subspace of a sum of Lindel¨of spaces and so would be paracompact. Pickwα∈W ∩Bα,α∈some stationaryE₀⊆ω₁. By Lemma 8, pick non-trivial continuaK_α such thatw_α ∈K_α ⊆B_α.

|W|=ℵ₁ <2^ℵ0, so pickq_α ∈K_α−W. LetM_α⁰ =Mα−W. W is closed, so M−W is open, so locally compact. M_α is hereditarily Lindel¨of, soM−W =S

α<ω1M_α⁰ is Type I.

20 / 26

Now we can finally prove:

Theorem 2 P

PP−

+CW+PPI+P22(ℵ₁)implies T5 manifolds of dimension greater than1are metrizable.

Proof.

Non-metrizableT5 Type I manifoldM includes a perfect pre-image ofω₁, and hence a copy W ofω₁. W must meet stationarily many Bα, else it would be a closed subspace of a sum of Lindel¨of spaces and so would be paracompact. Pickwα∈W ∩Bα,α∈some stationaryE₀⊆ω₁. By Lemma 8, pick non-trivial continuaK_α such thatw_α ∈K_α ⊆B_α. |W|=ℵ₁<2^ℵ0, so pickq_α ∈K_α−W.

LetM_α⁰ =Mα−W. W is closed, so M−W is open, so locally compact. M_α is hereditarily Lindel¨of, soM−W =S

α<ω1M_α⁰ is Type I.

Now we can finally prove:

Theorem 2 P

PP−

+CW+PPI+P22(ℵ₁)implies T5 manifolds of dimension greater than1are metrizable.

Proof.

Non-metrizableT5 Type I manifoldM includes a perfect pre-image ofω₁, and hence a copy W ofω₁. W must meet stationarily many Bα, else it would be a closed subspace of a sum of Lindel¨of spaces and so would be paracompact. Pickwα∈W ∩Bα,α∈some stationaryE₀⊆ω₁. By Lemma 8, pick non-trivial continuaK_α such thatw_α ∈K_α ⊆B_α. |W|=ℵ₁<2^ℵ0, so pickq_α ∈K_α−W. LetM_α⁰ =Mα−W.

W is closed, so M−W is open, so locally compact. M_α is hereditarily Lindel¨of, soM−W =S

α<ω1M_α⁰ is Type I.

20 / 26

Now we can finally prove:

Theorem 2 P

PP−

+CW+PPI+P22(ℵ₁)implies T5 manifolds of dimension greater than1are metrizable.

Proof.

Non-metrizableT5 Type I manifoldM includes a perfect pre-image ofω₁, and hence a copy W ofω₁. W must meet stationarily many Bα, else it would be a closed subspace of a sum of Lindel¨of spaces and so would be paracompact. Pickwα∈W ∩Bα,α∈some stationaryE₀⊆ω₁. By Lemma 8, pick non-trivial continuaK_α such thatw_α ∈K_α ⊆B_α. |W|=ℵ₁<2^ℵ0, so pickq_α ∈K_α−W. LetM_α⁰ =Mα−W. W is closed, so M−W is open, so locally compact.

M_α is hereditarily Lindel¨of, soM−W =S

α<ω1M_α⁰ is Type I.

Now we can finally prove:

Theorem 2 P

PP−

+CW+PPI+P22(ℵ₁)implies T5 manifolds of dimension greater than1are metrizable.

Proof.

Non-metrizableT5 Type I manifoldM includes a perfect pre-image ofω₁, and hence a copy W ofω₁. W must meet stationarily many Bα, else it would be a closed subspace of a sum of Lindel¨of spaces and so would be paracompact. Pickwα∈W ∩Bα,α∈some stationaryE₀⊆ω₁. By Lemma 8, pick non-trivial continuaK_α such thatw_α ∈K_α ⊆B_α. |W|=ℵ₁<2^ℵ0, so pickq_α ∈K_α−W. LetM_α⁰ =Mα−W. W is closed, so M−W is open, so locally compact. M_α is hereditarily Lindel¨of, so M−W =S

α<ω1M_α⁰ is Type I.

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Proof continued.

A Type I space is paracompact iff for (some) every canonical sequence{M_α}_α<ω

1, club many (M_α−M_α)’s are empty.

ThereforeM −W is not paracompact. ByP22(ℵ₁) and Lemma 4, it includes a perfect pre-imageQ of ω₁,Q = the closure of {q_α :α∈E₁} in M−W, for a stationary E₁⊆E₀.

Q is countably compact and hence closed in M. Let

f :M →[0,1], f(W) = 0,f(Q) = 1. Thenf(K_α) = [0,1] for each α∈E1. Forα’s inE1, recursively pickzα ∈Kα such that∀β < α, f(x_β)6=f(x_α). ByP₂₂(ℵ₁) and Lemmas 4, 11, there is a

stationaryE ⊆E₁ such thatZ ={z_α:α∈E} is a perfect pre-image ofω₁.

Proof continued.

A Type I space is paracompact iff for (some) every canonical sequence{M_α}_α<ω

1, club many (M_α−M_α)’s are empty.

ThereforeM −W is not paracompact.

By P22(ℵ₁) and Lemma 4, it includes a perfect pre-imageQ of ω₁,Q = the closure of {q_α :α∈E₁} in M−W, for a stationary E₁⊆E₀.

Q is countably compact and hence closed in M. Let

f :M →[0,1], f(W) = 0,f(Q) = 1. Thenf(K_α) = [0,1] for each α∈E1. Forα’s inE1, recursively pickzα ∈Kα such that∀β < α, f(x_β)6=f(x_α). ByP₂₂(ℵ₁) and Lemmas 4, 11, there is a

stationaryE ⊆E₁ such thatZ ={z_α:α∈E} is a perfect pre-image ofω₁.

21 / 26

Proof continued.

A Type I space is paracompact iff for (some) every canonical sequence{M_α}_α<ω

1, club many (M_α−M_α)’s are empty.

ThereforeM −W is not paracompact. By P₂₂(ℵ₁) and Lemma 4, it includes a perfect pre-imageQ of ω₁,Q = the closure of {q_α :α∈E₁} in M−W, for a stationary E₁⊆E₀.

Q is countably compact and hence closed in M. Let

f :M →[0,1], f(W) = 0,f(Q) = 1. Thenf(K_α) = [0,1] for each α∈E1. Forα’s inE1, recursively pickzα ∈Kα such that∀β < α, f(x_β)6=f(x_α). ByP₂₂(ℵ₁) and Lemmas 4, 11, there is a

stationaryE ⊆E₁ such thatZ ={z_α:α∈E} is a perfect pre-image ofω₁.

Proof continued.

A Type I space is paracompact iff for (some) every canonical sequence{M_α}_α<ω

1, club many (M_α−M_α)’s are empty.

ThereforeM −W is not paracompact. By P₂₂(ℵ₁) and Lemma 4, it includes a perfect pre-imageQ of ω₁,Q = the closure of {q_α :α∈E₁} in M−W, for a stationary E₁⊆E₀.

Q is countably compact and hence closed in M.

Let

f :M →[0,1], f(W) = 0,f(Q) = 1. Thenf(K_α) = [0,1] for each α∈E1. Forα’s inE1, recursively pickzα ∈Kα such that∀β < α, f(x_β)6=f(x_α). ByP₂₂(ℵ₁) and Lemmas 4, 11, there is a

stationaryE ⊆E₁ such thatZ ={z_α:α∈E} is a perfect pre-image ofω₁.

21 / 26

Proof continued.

A Type I space is paracompact iff for (some) every canonical sequence{M_α}_α<ω

1, club many (M_α−M_α)’s are empty.

ThereforeM −W is not paracompact. By P₂₂(ℵ₁) and Lemma 4, it includes a perfect pre-imageQ of ω₁,Q = the closure of {q_α :α∈E₁} in M−W, for a stationary E₁⊆E₀.

Q is countably compact and hence closed in M. Let

f :M →[0,1], f(W) = 0,f(Q) = 1. Thenf(K_α) = [0,1] for each α∈E1.

Forα’s inE1, recursively pickzα ∈Kα such that∀β < α, f(x_β)6=f(x_α). ByP₂₂(ℵ₁) and Lemmas 4, 11, there is a

stationaryE ⊆E₁ such thatZ ={z_α:α∈E} is a perfect pre-image ofω₁.

Proof continued.

A Type I space is paracompact iff for (some) every canonical sequence{M_α}_α<ω

1, club many (M_α−M_α)’s are empty.

ThereforeM −W is not paracompact. By P₂₂(ℵ₁) and Lemma 4, it includes a perfect pre-imageQ of ω₁,Q = the closure of {q_α :α∈E₁} in M−W, for a stationary E₁⊆E₀.

Q is countably compact and hence closed in M. Let

f :M →[0,1], f(W) = 0,f(Q) = 1. Thenf(K_α) = [0,1] for each α∈E1. Forα’s inE1, recursively pickzα ∈Kα such that∀β < α, f(x_β)6=f(x_α).

By P₂₂(ℵ₁) and Lemmas 4, 11, there is a stationaryE ⊆E₁ such thatZ ={z_α:α∈E} is a perfect pre-image ofω₁.

21 / 26

Proof continued.

A Type I space is paracompact iff for (some) every canonical sequence{M_α}_α<ω

1, club many (M_α−M_α)’s are empty.

ThereforeM −W is not paracompact. By P₂₂(ℵ₁) and Lemma 4, it includes a perfect pre-imageQ of ω₁,Q = the closure of {q_α :α∈E₁} in M−W, for a stationary E₁⊆E₀.

Q is countably compact and hence closed in M. Let

f :M →[0,1], f(W) = 0,f(Q) = 1. Thenf(K_α) = [0,1] for each α∈E1. Forα’s inE1, recursively pickzα ∈Kα such that∀β < α, f(x_β)6=f(x_α). ByP₂₂(ℵ₁) and Lemmas 4, 11, there is a

stationaryE ⊆E₁ such thatZ ={z_α :α∈E} is a perfect pre-image ofω₁.