[PPT] - Resolvability properties of certain topological spaces Istvn Juhsz PowerPoint Presentation

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Resolvability properties of certain topological spaces

István Juhász

Alfréd Rényi Institute of Mathematics

Sao Paulo, Brasil, August 2013

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 1 / 18

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resolvability

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 2 / 18

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resolvability

DEFINITION. (Hewitt, 1943, Pearson, 1963)

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 2 / 18

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resolvability

DEFINITION. (Hewitt, 1943, Pearson, 1963)

– A topological space X is κ-resolvable iff it has κ disjoint dense subsets.

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 2 / 18

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resolvability

DEFINITION. (Hewitt, 1943, Pearson, 1963)

– A topological space X is κ-resolvable iff it has κ disjoint dense subsets. (resolvable ≡ 2-resolvable)

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 2 / 18

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resolvability

DEFINITION. (Hewitt, 1943, Pearson, 1963)

– A topological space X is κ-resolvable iff it has κ disjoint dense subsets. (resolvable ≡ 2-resolvable) – X is maximally resolvable iff it is ∆(X)-resolvable,

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 2 / 18

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resolvability

DEFINITION. (Hewitt, 1943, Pearson, 1963)

– A topological space X is κ-resolvable iff it has κ disjoint dense subsets. (resolvable ≡ 2-resolvable) – X is maximally resolvable iff it is ∆(X)-resolvable, where ∆(X) = min{|G| : G = ∅ open in X} .

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 2 / 18

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resolvability

DEFINITION. (Hewitt, 1943, Pearson, 1963)

– A topological space X is κ-resolvable iff it has κ disjoint dense subsets. (resolvable ≡ 2-resolvable) – X is maximally resolvable iff it is ∆(X)-resolvable, where ∆(X) = min{|G| : G = ∅ open in X} . EXAMPLES:

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 2 / 18

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resolvability

DEFINITION. (Hewitt, 1943, Pearson, 1963)

– A topological space X is κ-resolvable iff it has κ disjoint dense subsets. (resolvable ≡ 2-resolvable) – X is maximally resolvable iff it is ∆(X)-resolvable, where ∆(X) = min{|G| : G = ∅ open in X} . EXAMPLES: – R is maximally resolvable.

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 2 / 18

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resolvability

DEFINITION. (Hewitt, 1943, Pearson, 1963)

– A topological space X is κ-resolvable iff it has κ disjoint dense subsets. (resolvable ≡ 2-resolvable) – X is maximally resolvable iff it is ∆(X)-resolvable, where ∆(X) = min{|G| : G = ∅ open in X} . EXAMPLES: – R is maximally resolvable. – Compact Hausdorff,

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 2 / 18

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resolvability

DEFINITION. (Hewitt, 1943, Pearson, 1963)

– A topological space X is κ-resolvable iff it has κ disjoint dense subsets. (resolvable ≡ 2-resolvable) – X is maximally resolvable iff it is ∆(X)-resolvable, where ∆(X) = min{|G| : G = ∅ open in X} . EXAMPLES: – R is maximally resolvable. – Compact Hausdorff, metric,

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 2 / 18

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resolvability

DEFINITION. (Hewitt, 1943, Pearson, 1963)

– A topological space X is κ-resolvable iff it has κ disjoint dense subsets. (resolvable ≡ 2-resolvable) – X is maximally resolvable iff it is ∆(X)-resolvable, where ∆(X) = min{|G| : G = ∅ open in X} . EXAMPLES: – R is maximally resolvable. – Compact Hausdorff, metric, and linearly ordered spaces are maximally resolvable.

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 2 / 18

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resolvability

DEFINITION. (Hewitt, 1943, Pearson, 1963)

– A topological space X is κ-resolvable iff it has κ disjoint dense subsets. (resolvable ≡ 2-resolvable) – X is maximally resolvable iff it is ∆(X)-resolvable, where ∆(X) = min{|G| : G = ∅ open in X} . EXAMPLES: – R is maximally resolvable. – Compact Hausdorff, metric, and linearly ordered spaces are maximally resolvable.

QUESTION. What happens if these properties are relaxed?

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 2 / 18

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countably compact spaces

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 3 / 18

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countably compact spaces

THEOREM. (Pytkeev, 2006)

Every crowded countably compact T3 space X is ω1-resolvable.

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 3 / 18

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countably compact spaces

THEOREM. (Pytkeev, 2006)

Every crowded countably compact T3 space X is ω1-resolvable.

NOTE. This fails for T2!

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 3 / 18

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countably compact spaces

THEOREM. (Pytkeev, 2006)

Every crowded countably compact T3 space X is ω1-resolvable.

NOTE. This fails for T2!

PROOF .(Not Pytkeev’s)

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 3 / 18

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countably compact spaces

THEOREM. (Pytkeev, 2006)

Every crowded countably compact T3 space X is ω1-resolvable.

NOTE. This fails for T2!

PROOF .(Not Pytkeev’s) Tkachenko (1979): If Y is countably compact T3 with ls(Y) ≤ ω then Y is scattered.

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 3 / 18

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countably compact spaces

THEOREM. (Pytkeev, 2006)

Every crowded countably compact T3 space X is ω1-resolvable.

NOTE. This fails for T2!

PROOF .(Not Pytkeev’s) Tkachenko (1979): If Y is countably compact T3 with ls(Y) ≤ ω then Y is scattered. But every open G ⊂ X includes a regular closed Y, hence ls(G) ≥ ls(Y) ≥ ω1.

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 3 / 18

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countably compact spaces

THEOREM. (Pytkeev, 2006)

Every crowded countably compact T3 space X is ω1-resolvable.

NOTE. This fails for T2!

PROOF .(Not Pytkeev’s) Tkachenko (1979): If Y is countably compact T3 with ls(Y) ≤ ω then Y is scattered. But every open G ⊂ X includes a regular closed Y, hence ls(G) ≥ ls(Y) ≥ ω1. So, any maximal disjoint family of dense left separated subsets of X must be uncountable.

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 3 / 18

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countably compact spaces

THEOREM. (Pytkeev, 2006)

Every crowded countably compact T3 space X is ω1-resolvable.

NOTE. This fails for T2!

PROOF .(Not Pytkeev’s) Tkachenko (1979): If Y is countably compact T3 with ls(Y) ≤ ω then Y is scattered. But every open G ⊂ X includes a regular closed Y, hence ls(G) ≥ ls(Y) ≥ ω1. So, any maximal disjoint family of dense left separated subsets of X must be uncountable.

PROBLEM.

Is every crowded countably compact T3 space X c-resolvable?

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 3 / 18

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countably compact spaces

THEOREM. (Pytkeev, 2006)

Every crowded countably compact T3 space X is ω1-resolvable.

NOTE. This fails for T2!

PROOF .(Not Pytkeev’s) Tkachenko (1979): If Y is countably compact T3 with ls(Y) ≤ ω then Y is scattered. But every open G ⊂ X includes a regular closed Y, hence ls(G) ≥ ls(Y) ≥ ω1. So, any maximal disjoint family of dense left separated subsets of X must be uncountable.

PROBLEM.

Is every crowded countably compact T3 space X c-resolvable? NOTE: ∆(X) ≥ c.

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 3 / 18

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countably compact spaces

THEOREM. (Pytkeev, 2006)

Every crowded countably compact T3 space X is ω1-resolvable.

NOTE. This fails for T2!

PROOF .(Not Pytkeev’s) Tkachenko (1979): If Y is countably compact T3 with ls(Y) ≤ ω then Y is scattered. But every open G ⊂ X includes a regular closed Y, hence ls(G) ≥ ls(Y) ≥ ω1. So, any maximal disjoint family of dense left separated subsets of X must be uncountable.

PROBLEM.

Is every crowded countably compact T3 space X c-resolvable? NOTE: ∆(X) ≥ c.

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 3 / 18

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Malychin’s problem

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 4 / 18

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Malychin’s problem

EXAMPLE. (Hewitt, ’43) There is a countable T3 space X that is

– crowded (i.e. ∆(X) = |X| = ℵ0) and

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 4 / 18

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Malychin’s problem

EXAMPLE. (Hewitt, ’43) There is a countable T3 space X that is

– crowded (i.e. ∆(X) = |X| = ℵ0) and – irresolvable( ≡ not 2-resolvable).

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 4 / 18

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Malychin’s problem

EXAMPLE. (Hewitt, ’43) There is a countable T3 space X that is

– crowded (i.e. ∆(X) = |X| = ℵ0) and – irresolvable( ≡ not 2-resolvable).

PROBLEM. (Malychin, 1995)

Is a Lindelöf T3 space X with ∆(X) > ω resolvable?

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 4 / 18

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Malychin’s problem

EXAMPLE. (Hewitt, ’43) There is a countable T3 space X that is

– crowded (i.e. ∆(X) = |X| = ℵ0) and – irresolvable( ≡ not 2-resolvable).

PROBLEM. (Malychin, 1995)

Is a Lindelöf T3 space X with ∆(X) > ω resolvable?

NOTE. Malychin constructed Lindelöf irresolvable Hausdorff (= T2)

spaces,

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 4 / 18

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Malychin’s problem

EXAMPLE. (Hewitt, ’43) There is a countable T3 space X that is

– crowded (i.e. ∆(X) = |X| = ℵ0) and – irresolvable( ≡ not 2-resolvable).

PROBLEM. (Malychin, 1995)

Is a Lindelöf T3 space X with ∆(X) > ω resolvable?

NOTE. Malychin constructed Lindelöf irresolvable Hausdorff (= T2)

spaces, and Pavlov Lindelöf irresolvable Uryson (= T2.5) spaces.

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 4 / 18

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Malychin’s problem

EXAMPLE. (Hewitt, ’43) There is a countable T3 space X that is

– crowded (i.e. ∆(X) = |X| = ℵ0) and – irresolvable( ≡ not 2-resolvable).

PROBLEM. (Malychin, 1995)

Is a Lindelöf T3 space X with ∆(X) > ω resolvable?

NOTE. Malychin constructed Lindelöf irresolvable Hausdorff (= T2)

spaces, and Pavlov Lindelöf irresolvable Uryson (= T2.5) spaces.

THEOREM. (Filatova, 2004)

YES, every Lindelöf T3 space X with ∆(X) > ω is 2-resolvable.

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 4 / 18

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Malychin’s problem

EXAMPLE. (Hewitt, ’43) There is a countable T3 space X that is

– crowded (i.e. ∆(X) = |X| = ℵ0) and – irresolvable( ≡ not 2-resolvable).

PROBLEM. (Malychin, 1995)

Is a Lindelöf T3 space X with ∆(X) > ω resolvable?

NOTE. Malychin constructed Lindelöf irresolvable Hausdorff (= T2)

spaces, and Pavlov Lindelöf irresolvable Uryson (= T2.5) spaces.

THEOREM. (Filatova, 2004)

YES, every Lindelöf T3 space X with ∆(X) > ω is 2-resolvable. This is the main result of her PhD thesis.

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 4 / 18

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Malychin’s problem

EXAMPLE. (Hewitt, ’43) There is a countable T3 space X that is

– crowded (i.e. ∆(X) = |X| = ℵ0) and – irresolvable( ≡ not 2-resolvable).

PROBLEM. (Malychin, 1995)

Is a Lindelöf T3 space X with ∆(X) > ω resolvable?

NOTE. Malychin constructed Lindelöf irresolvable Hausdorff (= T2)

spaces, and Pavlov Lindelöf irresolvable Uryson (= T2.5) spaces.

THEOREM. (Filatova, 2004)

YES, every Lindelöf T3 space X with ∆(X) > ω is 2-resolvable. This is the main result of her PhD thesis. It didn’t work for 3 !

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 4 / 18

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Pavlov’s theorems

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 5 / 18

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Pavlov’s theorems

s(X) = sup{|D| : D ⊂ X is discrete}

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 5 / 18

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Pavlov’s theorems

s(X) = sup{|D| : D ⊂ X is discrete} e(X) = sup{|D| : D ⊂ X is closed discrete}

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 5 / 18

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Pavlov’s theorems

s(X) = sup{|D| : D ⊂ X is discrete} e(X) = sup{|D| : D ⊂ X is closed discrete}

THEOREM. (Pavlov, 2002)

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 5 / 18

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Pavlov’s theorems

s(X) = sup{|D| : D ⊂ X is discrete} e(X) = sup{|D| : D ⊂ X is closed discrete}

THEOREM. (Pavlov, 2002)

(i) Any T2 space X with ∆(X) > s(X)+ is maximally resolvable.

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 5 / 18

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Pavlov’s theorems

s(X) = sup{|D| : D ⊂ X is discrete} e(X) = sup{|D| : D ⊂ X is closed discrete}

THEOREM. (Pavlov, 2002)

(i) Any T2 space X with ∆(X) > s(X)+ is maximally resolvable. (ii) Any T3 space X with ∆(X) > e(X)+ is ω-resolvable.

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 5 / 18

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Pavlov’s theorems

s(X) = sup{|D| : D ⊂ X is discrete} e(X) = sup{|D| : D ⊂ X is closed discrete}

THEOREM. (Pavlov, 2002)

(i) Any T2 space X with ∆(X) > s(X)+ is maximally resolvable. (ii) Any T3 space X with ∆(X) > e(X)+ is ω-resolvable.

THEOREM. (J-S-Sz, 2007)

Any space X with ∆(X) > s(X) is maximally resolvable.

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 5 / 18

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Pavlov’s theorems

s(X) = sup{|D| : D ⊂ X is discrete} e(X) = sup{|D| : D ⊂ X is closed discrete}

THEOREM. (Pavlov, 2002)

(i) Any T2 space X with ∆(X) > s(X)+ is maximally resolvable. (ii) Any T3 space X with ∆(X) > e(X)+ is ω-resolvable.

THEOREM. (J-S-Sz, 2007)

Any space X with ∆(X) > s(X) is maximally resolvable.

THEOREM. (J-S-Sz, 2012)

Any T3 space X with ∆(X) > e(X) is ω-resolvable.

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 5 / 18

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Pavlov’s theorems

s(X) = sup{|D| : D ⊂ X is discrete} e(X) = sup{|D| : D ⊂ X is closed discrete}

THEOREM. (Pavlov, 2002)

(i) Any T2 space X with ∆(X) > s(X)+ is maximally resolvable. (ii) Any T3 space X with ∆(X) > e(X)+ is ω-resolvable.

THEOREM. (J-S-Sz, 2007)

Any space X with ∆(X) > s(X) is maximally resolvable.

THEOREM. (J-S-Sz, 2012)

Any T3 space X with ∆(X) > e(X) is ω-resolvable. In particular, every Lindelöf T3 space X with ∆(X) > ω is ω-resolvable.

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 5 / 18

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Pavlov’s theorems

s(X) = sup{|D| : D ⊂ X is discrete} e(X) = sup{|D| : D ⊂ X is closed discrete}

THEOREM. (Pavlov, 2002)

(i) Any T2 space X with ∆(X) > s(X)+ is maximally resolvable. (ii) Any T3 space X with ∆(X) > e(X)+ is ω-resolvable.

THEOREM. (J-S-Sz, 2007)

Any space X with ∆(X) > s(X) is maximally resolvable.

THEOREM. (J-S-Sz, 2012)

Any T3 space X with ∆(X) > e(X) is ω-resolvable. In particular, every Lindelöf T3 space X with ∆(X) > ω is ω-resolvable.

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 5 / 18

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J-S-Sz

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 6 / 18

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J-S-Sz

THEOREM. (J-S-Sz, 2007)

If ∆(X) ≥ κ = cf(κ) > ω and X has no discrete subset of size κ then X is κ-resolvable.

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 6 / 18

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J-S-Sz

THEOREM. (J-S-Sz, 2007)

If ∆(X) ≥ κ = cf(κ) > ω and X has no discrete subset of size κ then X is κ-resolvable.

THEOREM. (J-S-Sz, 2012)

If X is T3, ∆(X) ≥ κ = cf(κ) > ω and X has no closed discrete subset

f size κ then X is ω-resolvable.

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 6 / 18

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J-S-Sz

THEOREM. (J-S-Sz, 2007)

If ∆(X) ≥ κ = cf(κ) > ω and X has no discrete subset of size κ then X is κ-resolvable.

THEOREM. (J-S-Sz, 2012)

If X is T3, ∆(X) ≥ κ = cf(κ) > ω and X has no closed discrete subset

f size κ then X is ω-resolvable.
NOTE. For ∆(X) > ω regular these suffice.

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 6 / 18

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J-S-Sz

THEOREM. (J-S-Sz, 2007)

If ∆(X) ≥ κ = cf(κ) > ω and X has no discrete subset of size κ then X is κ-resolvable.

THEOREM. (J-S-Sz, 2012)

If X is T3, ∆(X) ≥ κ = cf(κ) > ω and X has no closed discrete subset

f size κ then X is ω-resolvable.
NOTE. For ∆(X) > ω regular these suffice. If ∆(X) = λ is singular,

we need something extra.

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 6 / 18

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J-S-Sz

THEOREM. (J-S-Sz, 2007)

If ∆(X) ≥ κ = cf(κ) > ω and X has no discrete subset of size κ then X is κ-resolvable.

THEOREM. (J-S-Sz, 2012)

If X is T3, ∆(X) ≥ κ = cf(κ) > ω and X has no closed discrete subset

f size κ then X is ω-resolvable.
NOTE. For ∆(X) > ω regular these suffice. If ∆(X) = λ is singular,

we need something extra. For ∆(X) = λ > s(X) we automatically get that X is < λ-resolvable.

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 6 / 18

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J-S-Sz

THEOREM. (J-S-Sz, 2007)

If ∆(X) ≥ κ = cf(κ) > ω and X has no discrete subset of size κ then X is κ-resolvable.

THEOREM. (J-S-Sz, 2012)

If X is T3, ∆(X) ≥ κ = cf(κ) > ω and X has no closed discrete subset

f size κ then X is ω-resolvable.
NOTE. For ∆(X) > ω regular these suffice. If ∆(X) = λ is singular,

we need something extra. For ∆(X) = λ > s(X) we automatically get that X is < λ-resolvable. But now ∆(X) = λ > s(X)+, so we may use Pavlov’s Thm (i).

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 6 / 18

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J-S-Sz

THEOREM. (J-S-Sz, 2007)

If ∆(X) ≥ κ = cf(κ) > ω and X has no discrete subset of size κ then X is κ-resolvable.

THEOREM. (J-S-Sz, 2012)

If X is T3, ∆(X) ≥ κ = cf(κ) > ω and X has no closed discrete subset

f size κ then X is ω-resolvable.
NOTE. For ∆(X) > ω regular these suffice. If ∆(X) = λ is singular,

we need something extra. For ∆(X) = λ > s(X) we automatically get that X is < λ-resolvable. But now ∆(X) = λ > s(X)+, so we may use Pavlov’s Thm (i). For ∆(X) = λ > e(X)+ we may use Pavlov’s Thm (ii).

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 6 / 18

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< λ-resolvable

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 7 / 18

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< λ-resolvable

THEOREM. (J-S-Sz, 2006)

For any κ ≥ λ = cf(λ) > ω there is a dense X ⊂ D(2)2κ with ∆(X) = κ that is < λ-resolvable but not λ-resolvable.

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 7 / 18

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< λ-resolvable

THEOREM. (J-S-Sz, 2006)

For any κ ≥ λ = cf(λ) > ω there is a dense X ⊂ D(2)2κ with ∆(X) = κ that is < λ-resolvable but not λ-resolvable.

NOTE. This solved a problem of Ceder and Pearson from 1967.

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 7 / 18

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< λ-resolvable

THEOREM. (J-S-Sz, 2006)

For any κ ≥ λ = cf(λ) > ω there is a dense X ⊂ D(2)2κ with ∆(X) = κ that is < λ-resolvable but not λ-resolvable.

NOTE. This solved a problem of Ceder and Pearson from 1967. We

used the general method of constructing D-forced spaces.

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 7 / 18

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< λ-resolvable

THEOREM. (J-S-Sz, 2006)

For any κ ≥ λ = cf(λ) > ω there is a dense X ⊂ D(2)2κ with ∆(X) = κ that is < λ-resolvable but not λ-resolvable.

NOTE. This solved a problem of Ceder and Pearson from 1967. We

used the general method of constructing D-forced spaces.

THEOREM. (Illanes, Baskara Rao)

If cf(λ) = ω then every < λ-resolvable space is λ-resolvable.

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 7 / 18

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< λ-resolvable

THEOREM. (J-S-Sz, 2006)

For any κ ≥ λ = cf(λ) > ω there is a dense X ⊂ D(2)2κ with ∆(X) = κ that is < λ-resolvable but not λ-resolvable.

NOTE. This solved a problem of Ceder and Pearson from 1967. We

used the general method of constructing D-forced spaces.

THEOREM. (Illanes, Baskara Rao)

If cf(λ) = ω then every < λ-resolvable space is λ-resolvable.

PROBLEM.

Is this true for each singular λ?

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 7 / 18

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< λ-resolvable

THEOREM. (J-S-Sz, 2006)

For any κ ≥ λ = cf(λ) > ω there is a dense X ⊂ D(2)2κ with ∆(X) = κ that is < λ-resolvable but not λ-resolvable.

NOTE. This solved a problem of Ceder and Pearson from 1967. We

used the general method of constructing D-forced spaces.

THEOREM. (Illanes, Baskara Rao)

If cf(λ) = ω then every < λ-resolvable space is λ-resolvable.

PROBLEM.

Is this true for each singular λ? How about λ = ℵω1?

István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 7 / 18

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< λ-resolvable

THEOREM. (J-S-Sz, 2006)

For any κ ≥ λ = cf(λ) > ω there is a dense X ⊂ D(2)2κ with ∆(X) = κ that is < λ-resolvable but not λ-resolvable.

NOTE. This solved a problem of Ceder and Pearson from 1967. We

used the general method of constructing D-forced spaces.

THEOREM. (Illanes, Baskara Rao)

If cf(λ) = ω then every < λ-resolvable space is λ-resolvable.

PROBLEM.

Is this true for each singular λ? How about λ = ℵω1?

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monotone normality

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monotone normality

DEFINITION.

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monotone normality

DEFINITION. The space X is monotonically normal ( MN ) iff it is T1 (i.e. all singletons are closed)

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monotone normality

DEFINITION. The space X is monotonically normal ( MN ) iff it is T1 (i.e. all singletons are closed) and it has a monotone normality operator H that "halves" neighbourhoods :

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monotone normality

DEFINITION. The space X is monotonically normal ( MN ) iff it is T1 (i.e. all singletons are closed) and it has a monotone normality operator H that "halves" neighbourhoods : H assigns to every x, U, with x ∈ U open, an open set H(x, U) s. t.

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monotone normality

DEFINITION. The space X is monotonically normal ( MN ) iff it is T1 (i.e. all singletons are closed) and it has a monotone normality operator H that "halves" neighbourhoods : H assigns to every x, U, with x ∈ U open, an open set H(x, U) s. t. (i) x ∈ H(x, U) ⊂ U ,

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monotone normality

DEFINITION. The space X is monotonically normal ( MN ) iff it is T1 (i.e. all singletons are closed) and it has a monotone normality operator H that "halves" neighbourhoods : H assigns to every x, U, with x ∈ U open, an open set H(x, U) s. t. (i) x ∈ H(x, U) ⊂ U , and (ii) if H(x, U) ∩ H(y, V) = ∅ then x ∈ V or y ∈ U .

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monotone normality

DEFINITION. The space X is monotonically normal ( MN ) iff it is T1 (i.e. all singletons are closed) and it has a monotone normality operator H that "halves" neighbourhoods : H assigns to every x, U, with x ∈ U open, an open set H(x, U) s. t. (i) x ∈ H(x, U) ⊂ U , and (ii) if H(x, U) ∩ H(y, V) = ∅ then x ∈ V or y ∈ U .

FACT. Metric spaces

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monotone normality

DEFINITION. The space X is monotonically normal ( MN ) iff it is T1 (i.e. all singletons are closed) and it has a monotone normality operator H that "halves" neighbourhoods : H assigns to every x, U, with x ∈ U open, an open set H(x, U) s. t. (i) x ∈ H(x, U) ⊂ U , and (ii) if H(x, U) ∩ H(y, V) = ∅ then x ∈ V or y ∈ U .

FACT. Metric spaces and linearly ordered spaces are MN.

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monotone normality

DEFINITION. The space X is monotonically normal ( MN ) iff it is T1 (i.e. all singletons are closed) and it has a monotone normality operator H that "halves" neighbourhoods : H assigns to every x, U, with x ∈ U open, an open set H(x, U) s. t. (i) x ∈ H(x, U) ⊂ U , and (ii) if H(x, U) ∩ H(y, V) = ∅ then x ∈ V or y ∈ U .

FACT. Metric spaces and linearly ordered spaces are MN.
QUESTION. Are MN spaces maximally resolvable?

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SD spaces

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SD spaces

DEFINITION.

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SD spaces

DEFINITION. (i) D ⊂ X is strongly discrete if there are pairwise disjoint

pen sets {Ux : x ∈ D} with x ∈ Ux for x ∈ D.

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SD spaces

DEFINITION. (i) D ⊂ X is strongly discrete if there are pairwise disjoint

pen sets {Ux : x ∈ D} with x ∈ Ux for x ∈ D.

EXAMPLE: Countable discrete sets in T3 spaces are SD.

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SD spaces

DEFINITION. (i) D ⊂ X is strongly discrete if there are pairwise disjoint

pen sets {Ux : x ∈ D} with x ∈ Ux for x ∈ D.

EXAMPLE: Countable discrete sets in T3 spaces are SD. (ii) X is an SD space if every non-isolated point x ∈ X is an SD limit.

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SD spaces

DEFINITION. (i) D ⊂ X is strongly discrete if there are pairwise disjoint

pen sets {Ux : x ∈ D} with x ∈ Ux for x ∈ D.

EXAMPLE: Countable discrete sets in T3 spaces are SD. (ii) X is an SD space if every non-isolated point x ∈ X is an SD limit.

THEOREM. (Sharma and Sharma, 1988)

Every T1 crowded SD space is ω-resolvable.

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SD spaces

DEFINITION. (i) D ⊂ X is strongly discrete if there are pairwise disjoint

pen sets {Ux : x ∈ D} with x ∈ Ux for x ∈ D.

EXAMPLE: Countable discrete sets in T3 spaces are SD. (ii) X is an SD space if every non-isolated point x ∈ X is an SD limit.

THEOREM. (Sharma and Sharma, 1988)

Every T1 crowded SD space is ω-resolvable.

THEOREM. (DTTW, 2002)

MN spaces are SD, hence crowded MN spaces are ω-resolvable.

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SD spaces

DEFINITION. (i) D ⊂ X is strongly discrete if there are pairwise disjoint

pen sets {Ux : x ∈ D} with x ∈ Ux for x ∈ D.

EXAMPLE: Countable discrete sets in T3 spaces are SD. (ii) X is an SD space if every non-isolated point x ∈ X is an SD limit.

THEOREM. (Sharma and Sharma, 1988)

Every T1 crowded SD space is ω-resolvable.

THEOREM. (DTTW, 2002)

MN spaces are SD, hence crowded MN spaces are ω-resolvable.

PROBLEM. (Ceder and Pearson, 1967)

Are ω-resolvable spaces maximally resolvable?

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[J-S-Sz]

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[J-S-Sz]

[J-S-Sz] ≡ I. JUHÁSZ, L. SOUKUP AND Z. SZENTMIKLÓSSY, Resolvability and monotone normality, Israel J. Math., 166 (2008),

no. 1, pp. 1–16.

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[J-S-Sz]

[J-S-Sz] ≡ I. JUHÁSZ, L. SOUKUP AND Z. SZENTMIKLÓSSY, Resolvability and monotone normality, Israel J. Math., 166 (2008),

no. 1, pp. 1–16.
DEFINITION. X is a DSD space if every dense subspace of X is SD.

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[J-S-Sz]

[J-S-Sz] ≡ I. JUHÁSZ, L. SOUKUP AND Z. SZENTMIKLÓSSY, Resolvability and monotone normality, Israel J. Math., 166 (2008),

no. 1, pp. 1–16.
DEFINITION. X is a DSD space if every dense subspace of X is SD.

Clearly, MN spaces are DSD.

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[J-S-Sz]

[J-S-Sz] ≡ I. JUHÁSZ, L. SOUKUP AND Z. SZENTMIKLÓSSY, Resolvability and monotone normality, Israel J. Math., 166 (2008),

no. 1, pp. 1–16.
DEFINITION. X is a DSD space if every dense subspace of X is SD.

Clearly, MN spaces are DSD.

Main results of [J-S-Sz]

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[J-S-Sz]

[J-S-Sz] ≡ I. JUHÁSZ, L. SOUKUP AND Z. SZENTMIKLÓSSY, Resolvability and monotone normality, Israel J. Math., 166 (2008),

no. 1, pp. 1–16.
DEFINITION. X is a DSD space if every dense subspace of X is SD.

Clearly, MN spaces are DSD.

Main results of [J-S-Sz]

– If κ is measurable then there is a MN space X with ∆(X) = κ that is ω1-irrresolvable.

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[J-S-Sz]

[J-S-Sz] ≡ I. JUHÁSZ, L. SOUKUP AND Z. SZENTMIKLÓSSY, Resolvability and monotone normality, Israel J. Math., 166 (2008),

no. 1, pp. 1–16.
DEFINITION. X is a DSD space if every dense subspace of X is SD.

Clearly, MN spaces are DSD.

Main results of [J-S-Sz]

– If κ is measurable then there is a MN space X with ∆(X) = κ that is ω1-irrresolvable. – If X is DSD with |X| < ℵω then X is maximally resolvable.

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[J-S-Sz]

[J-S-Sz] ≡ I. JUHÁSZ, L. SOUKUP AND Z. SZENTMIKLÓSSY, Resolvability and monotone normality, Israel J. Math., 166 (2008),

no. 1, pp. 1–16.
DEFINITION. X is a DSD space if every dense subspace of X is SD.

Clearly, MN spaces are DSD.

Main results of [J-S-Sz]

– If κ is measurable then there is a MN space X with ∆(X) = κ that is ω1-irrresolvable. – If X is DSD with |X| < ℵω then X is maximally resolvable. – From a supercompact cardinal, it is consistent to have a MN space X with |X| = ∆(X) = ℵω that is ω2-irresolvable.

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[J-S-Sz]

[J-S-Sz] ≡ I. JUHÁSZ, L. SOUKUP AND Z. SZENTMIKLÓSSY, Resolvability and monotone normality, Israel J. Math., 166 (2008),

no. 1, pp. 1–16.
DEFINITION. X is a DSD space if every dense subspace of X is SD.

Clearly, MN spaces are DSD.

Main results of [J-S-Sz]

– If κ is measurable then there is a MN space X with ∆(X) = κ that is ω1-irrresolvable. – If X is DSD with |X| < ℵω then X is maximally resolvable. – From a supercompact cardinal, it is consistent to have a MN space X with |X| = ∆(X) = ℵω that is ω2-irresolvable. This left a number of questions open.

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decomposability of ultrafilters

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decomposability of ultrafilters

DEFINITION.

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decomposability of ultrafilters

DEFINITION. – An ultrafilter F is µ-descendingly complete iff for any descending µ-sequence {Aα : α < µ} ⊂ F we have {Aα : α < µ} ∈ F.

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decomposability of ultrafilters

DEFINITION. – An ultrafilter F is µ-descendingly complete iff for any descending µ-sequence {Aα : α < µ} ⊂ F we have {Aα : α < µ} ∈ F. µ-descendingly incomplete is (now) called µ-decomposable.

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decomposability of ultrafilters

DEFINITION. – An ultrafilter F is µ-descendingly complete iff for any descending µ-sequence {Aα : α < µ} ⊂ F we have {Aα : α < µ} ∈ F. µ-descendingly incomplete is (now) called µ-decomposable. – ∆(F) = min{|A| : A ∈ F}.

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decomposability of ultrafilters

DEFINITION. – An ultrafilter F is µ-descendingly complete iff for any descending µ-sequence {Aα : α < µ} ⊂ F we have {Aα : α < µ} ∈ F. µ-descendingly incomplete is (now) called µ-decomposable. – ∆(F) = min{|A| : A ∈ F}. – F is maximally decomposable iff it is µ-decomposable for all (infinite) µ ≤ ∆(F).

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decomposability of ultrafilters

DEFINITION. – An ultrafilter F is µ-descendingly complete iff for any descending µ-sequence {Aα : α < µ} ⊂ F we have {Aα : α < µ} ∈ F. µ-descendingly incomplete is (now) called µ-decomposable. – ∆(F) = min{|A| : A ∈ F}. – F is maximally decomposable iff it is µ-decomposable for all (infinite) µ ≤ ∆(F). FACTS.

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decomposability of ultrafilters

DEFINITION. – An ultrafilter F is µ-descendingly complete iff for any descending µ-sequence {Aα : α < µ} ⊂ F we have {Aα : α < µ} ∈ F. µ-descendingly incomplete is (now) called µ-decomposable. – ∆(F) = min{|A| : A ∈ F}. – F is maximally decomposable iff it is µ-decomposable for all (infinite) µ ≤ ∆(F). FACTS. – Any "measure" is countably complete, hence ω-indecomposable.

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decomposability of ultrafilters

DEFINITION. – An ultrafilter F is µ-descendingly complete iff for any descending µ-sequence {Aα : α < µ} ⊂ F we have {Aα : α < µ} ∈ F. µ-descendingly incomplete is (now) called µ-decomposable. – ∆(F) = min{|A| : A ∈ F}. – F is maximally decomposable iff it is µ-decomposable for all (infinite) µ ≤ ∆(F). FACTS. – Any "measure" is countably complete, hence ω-indecomposable. – [Donder, 1988] If there is a not maximally decomposable ultrafilter then there is a measurable cardinal in some inner model.

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decomposability of ultrafilters

DEFINITION. – An ultrafilter F is µ-descendingly complete iff for any descending µ-sequence {Aα : α < µ} ⊂ F we have {Aα : α < µ} ∈ F. µ-descendingly incomplete is (now) called µ-decomposable. – ∆(F) = min{|A| : A ∈ F}. – F is maximally decomposable iff it is µ-decomposable for all (infinite) µ ≤ ∆(F). FACTS. – Any "measure" is countably complete, hence ω-indecomposable. – [Donder, 1988] If there is a not maximally decomposable ultrafilter then there is a measurable cardinal in some inner model. – [Kunen - Prikry, 1971] Every ultrafilter F with ∆(F) < ℵω is maximally decomposable.

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[J-M]

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[J-M]

[J-M] ≡ I. JUHÁSZ AND M. MAGIDOR, On the maximal resolvability of monotonically normal spaces, Israel J. Math, 192 (2012), 637-666.

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[J-M]

[J-M] ≡ I. JUHÁSZ AND M. MAGIDOR, On the maximal resolvability of monotonically normal spaces, Israel J. Math, 192 (2012), 637-666.

Main results of [J-M]

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SLIDE 99

[J-M]

[J-M] ≡ I. JUHÁSZ AND M. MAGIDOR, On the maximal resolvability of monotonically normal spaces, Israel J. Math, 192 (2012), 637-666.

Main results of [J-M]

(1) TFAEV

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SLIDE 100

[J-M]

[J-M] ≡ I. JUHÁSZ AND M. MAGIDOR, On the maximal resolvability of monotonically normal spaces, Israel J. Math, 192 (2012), 637-666.

Main results of [J-M]

(1) TFAEV – Every DSD space (of cardinality < κ) is maximally resolvable.

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[J-M]

[J-M] ≡ I. JUHÁSZ AND M. MAGIDOR, On the maximal resolvability of monotonically normal spaces, Israel J. Math, 192 (2012), 637-666.

Main results of [J-M]

(1) TFAEV – Every DSD space (of cardinality < κ) is maximally resolvable. – Every MN space (of cardinality < κ) is maximally resolvable.

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SLIDE 102

[J-M]

[J-M] ≡ I. JUHÁSZ AND M. MAGIDOR, On the maximal resolvability of monotonically normal spaces, Israel J. Math, 192 (2012), 637-666.

Main results of [J-M]

(1) TFAEV – Every DSD space (of cardinality < κ) is maximally resolvable. – Every MN space (of cardinality < κ) is maximally resolvable. – Every ultrafilter F (with ∆(F) < κ) is maximally decomposable.

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SLIDE 103

[J-M]

[J-M] ≡ I. JUHÁSZ AND M. MAGIDOR, On the maximal resolvability of monotonically normal spaces, Israel J. Math, 192 (2012), 637-666.

Main results of [J-M]

(1) TFAEV – Every DSD space (of cardinality < κ) is maximally resolvable. – Every MN space (of cardinality < κ) is maximally resolvable. – Every ultrafilter F (with ∆(F) < κ) is maximally decomposable. (2) TFAEC

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SLIDE 104

[J-M]

[J-M] ≡ I. JUHÁSZ AND M. MAGIDOR, On the maximal resolvability of monotonically normal spaces, Israel J. Math, 192 (2012), 637-666.

Main results of [J-M]

(1) TFAEV – Every DSD space (of cardinality < κ) is maximally resolvable. – Every MN space (of cardinality < κ) is maximally resolvable. – Every ultrafilter F (with ∆(F) < κ) is maximally decomposable. (2) TFAEC – There is a measurable cardinal.

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SLIDE 105

[J-M]

[J-M] ≡ I. JUHÁSZ AND M. MAGIDOR, On the maximal resolvability of monotonically normal spaces, Israel J. Math, 192 (2012), 637-666.

Main results of [J-M]

(1) TFAEV – Every DSD space (of cardinality < κ) is maximally resolvable. – Every MN space (of cardinality < κ) is maximally resolvable. – Every ultrafilter F (with ∆(F) < κ) is maximally decomposable. (2) TFAEC – There is a measurable cardinal. – There is a MN space that is not maximally resolvable.

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[J-M]

[J-M] ≡ I. JUHÁSZ AND M. MAGIDOR, On the maximal resolvability of monotonically normal spaces, Israel J. Math, 192 (2012), 637-666.

Main results of [J-M]

(1) TFAEV – Every DSD space (of cardinality < κ) is maximally resolvable. – Every MN space (of cardinality < κ) is maximally resolvable. – Every ultrafilter F (with ∆(F) < κ) is maximally decomposable. (2) TFAEC – There is a measurable cardinal. – There is a MN space that is not maximally resolvable. – There is a MN space X with |X| = ∆(X) = ℵω that is ω1-irresolvable.

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filtration spaces

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filtration spaces

DEFINITION.

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filtration spaces

DEFINITION. – F is a filtration if dom(F) = T is an infinitely branching tree

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filtration spaces

DEFINITION. – F is a filtration if dom(F) = T is an infinitely branching tree (of height ω)

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filtration spaces

DEFINITION. – F is a filtration if dom(F) = T is an infinitely branching tree (of height ω) and, for each t ∈ T, F(t) is a filter on S(t) that contains all co-finite subsets of S(t).

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filtration spaces

DEFINITION. – F is a filtration if dom(F) = T is an infinitely branching tree (of height ω) and, for each t ∈ T, F(t) is a filter on S(t) that contains all co-finite subsets of S(t). – The topology τF on T: For G ⊂ T , G ∈ τF iff t ∈ G ⇒ G ∩ S(t) ∈ F(t) ,

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filtration spaces

DEFINITION. – F is a filtration if dom(F) = T is an infinitely branching tree (of height ω) and, for each t ∈ T, F(t) is a filter on S(t) that contains all co-finite subsets of S(t). – The topology τF on T: For G ⊂ T , G ∈ τF iff t ∈ G ⇒ G ∩ S(t) ∈ F(t) , – X(F) = T, τF is called a filtration space.

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filtration spaces

DEFINITION. – F is a filtration if dom(F) = T is an infinitely branching tree (of height ω) and, for each t ∈ T, F(t) is a filter on S(t) that contains all co-finite subsets of S(t). – The topology τF on T: For G ⊂ T , G ∈ τF iff t ∈ G ⇒ G ∩ S(t) ∈ F(t) , – X(F) = T, τF is called a filtration space.

FACT. [J-S-Sz] Every filtration space X(F) is MN.

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filtration spaces

DEFINITION. – F is a filtration if dom(F) = T is an infinitely branching tree (of height ω) and, for each t ∈ T, F(t) is a filter on S(t) that contains all co-finite subsets of S(t). – The topology τF on T: For G ⊂ T , G ∈ τF iff t ∈ G ⇒ G ∩ S(t) ∈ F(t) , – X(F) = T, τF is called a filtration space.

FACT. [J-S-Sz] Every filtration space X(F) is MN.

Moreover, filtration spaces determine the resolvability behavior

f all MN (or DSD) spaces.

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irresolvability of filtration spaces

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irresolvability of filtration spaces

THEOREM. [J-S-Sz]

If F is an ultrafiltration and µ ≥ ω is a regular cardinal s.t. F(t) is µ-descendingly complete for all t ∈ T = dom(F),

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irresolvability of filtration spaces

THEOREM. [J-S-Sz]

If F is an ultrafiltration and µ ≥ ω is a regular cardinal s.t. F(t) is µ-descendingly complete for all t ∈ T = dom(F), then X(F) is hereditarily µ+-irresolvable.

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irresolvability of filtration spaces

THEOREM. [J-S-Sz]

If F is an ultrafiltration and µ ≥ ω is a regular cardinal s.t. F(t) is µ-descendingly complete for all t ∈ T = dom(F), then X(F) is hereditarily µ+-irresolvable.

COROLLARY. [J-S-Sz]

If F ∈ un(κ) is a measure and F(t) = F for all t ∈ dom(F) = κ<ω then X(F) is hereditarily ω1-irresolvable.

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λ-filtrations

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λ-filtrations

DEFINITION. [J-M]

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λ-filtrations

DEFINITION. [J-M] F is a λ-filtration if

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λ-filtrations

DEFINITION. [J-M] F is a λ-filtration if

– T = dom(F) ⊂ λ<ω,

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λ-filtrations

DEFINITION. [J-M] F is a λ-filtration if

– T = dom(F) ⊂ λ<ω, – for each t ∈ T there is ω ≤ µt ≤ λ s.t. S(t) = {tα : α < µt} and F(t) ∈ un(µt) ,

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λ-filtrations

DEFINITION. [J-M] F is a λ-filtration if

– T = dom(F) ⊂ λ<ω, – for each t ∈ T there is ω ≤ µt ≤ λ s.t. S(t) = {tα : α < µt} and F(t) ∈ un(µt) , – moreover, for any µ < λ and t ∈ T: {α : µtα > µ} ∈ F(t) .

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λ-filtrations

DEFINITION. [J-M] F is a λ-filtration if

– T = dom(F) ⊂ λ<ω, – for each t ∈ T there is ω ≤ µt ≤ λ s.t. S(t) = {tα : α < µt} and F(t) ∈ un(µt) , – moreover, for any µ < λ and t ∈ T: {α : µtα > µ} ∈ F(t) .

NOTE. If F is a λ-filtration then |X(F)| = ∆(X(F)) = λ.

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λ-filtrations

DEFINITION. [J-M] F is a λ-filtration if

– T = dom(F) ⊂ λ<ω, – for each t ∈ T there is ω ≤ µt ≤ λ s.t. S(t) = {tα : α < µt} and F(t) ∈ un(µt) , – moreover, for any µ < λ and t ∈ T: {α : µtα > µ} ∈ F(t) .

NOTE. If F is a λ-filtration then |X(F)| = ∆(X(F)) = λ.

– The λ-filtration F is full if dom(F) = λ<ω, i.e. µt = λ for all t ∈ λ<ω.

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λ-filtrations

DEFINITION. [J-M] F is a λ-filtration if

– T = dom(F) ⊂ λ<ω, – for each t ∈ T there is ω ≤ µt ≤ λ s.t. S(t) = {tα : α < µt} and F(t) ∈ un(µt) , – moreover, for any µ < λ and t ∈ T: {α : µtα > µ} ∈ F(t) .

NOTE. If F is a λ-filtration then |X(F)| = ∆(X(F)) = λ.

– The λ-filtration F is full if dom(F) = λ<ω, i.e. µt = λ for all t ∈ λ<ω. Full λ-filtrations were considered in [J-S-Sz].

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reduction results

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reduction results

THEOREM [J-S-Sz]

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reduction results

THEOREM [J-S-Sz]

For κ ≤ λ = cf(λ), TFAEV

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reduction results

THEOREM [J-S-Sz]

For κ ≤ λ = cf(λ), TFAEV – Every DSD space X with |X| = ∆(X) = λ is κ-resolvable.

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reduction results

THEOREM [J-S-Sz]

For κ ≤ λ = cf(λ), TFAEV – Every DSD space X with |X| = ∆(X) = λ is κ-resolvable. – Every MN space X with |X| = ∆(X) = λ is κ-resolvable.

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reduction results

THEOREM [J-S-Sz]

For κ ≤ λ = cf(λ), TFAEV – Every DSD space X with |X| = ∆(X) = λ is κ-resolvable. – Every MN space X with |X| = ∆(X) = λ is κ-resolvable. – For every full λ-filtration F, the space X(F) is κ-resolvable.

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reduction results

THEOREM [J-S-Sz]

For κ ≤ λ = cf(λ), TFAEV – Every DSD space X with |X| = ∆(X) = λ is κ-resolvable. – Every MN space X with |X| = ∆(X) = λ is κ-resolvable. – For every full λ-filtration F, the space X(F) is κ-resolvable.

THEOREM [J-M]

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reduction results

THEOREM [J-S-Sz]

For κ ≤ λ = cf(λ), TFAEV – Every DSD space X with |X| = ∆(X) = λ is κ-resolvable. – Every MN space X with |X| = ∆(X) = λ is κ-resolvable. – For every full λ-filtration F, the space X(F) is κ-resolvable.

THEOREM [J-M]

For λ singular and cf(λ)+ < κ ≤ λ, TFAEV

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reduction results

THEOREM [J-S-Sz]

For κ ≤ λ = cf(λ), TFAEV – Every DSD space X with |X| = ∆(X) = λ is κ-resolvable. – Every MN space X with |X| = ∆(X) = λ is κ-resolvable. – For every full λ-filtration F, the space X(F) is κ-resolvable.

THEOREM [J-M]

For λ singular and cf(λ)+ < κ ≤ λ, TFAEV – Every DSD space X with |X| = ∆(X) = λ is κ-resolvable.

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reduction results

THEOREM [J-S-Sz]

For κ ≤ λ = cf(λ), TFAEV – Every DSD space X with |X| = ∆(X) = λ is κ-resolvable. – Every MN space X with |X| = ∆(X) = λ is κ-resolvable. – For every full λ-filtration F, the space X(F) is κ-resolvable.

THEOREM [J-M]

For λ singular and cf(λ)+ < κ ≤ λ, TFAEV – Every DSD space X with |X| = ∆(X) = λ is κ-resolvable. – Every MN space X with |X| = ∆(X) = λ is κ-resolvable.

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reduction results

THEOREM [J-S-Sz]

For κ ≤ λ = cf(λ), TFAEV – Every DSD space X with |X| = ∆(X) = λ is κ-resolvable. – Every MN space X with |X| = ∆(X) = λ is κ-resolvable. – For every full λ-filtration F, the space X(F) is κ-resolvable.

THEOREM [J-M]

For λ singular and cf(λ)+ < κ ≤ λ, TFAEV – Every DSD space X with |X| = ∆(X) = λ is κ-resolvable. – Every MN space X with |X| = ∆(X) = λ is κ-resolvable. – For every λ-filtration F, the space X(F) is κ-resolvable.

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reduction results

THEOREM [J-S-Sz]

For κ ≤ λ = cf(λ), TFAEV – Every DSD space X with |X| = ∆(X) = λ is κ-resolvable. – Every MN space X with |X| = ∆(X) = λ is κ-resolvable. – For every full λ-filtration F, the space X(F) is κ-resolvable.

THEOREM [J-M]

For λ singular and cf(λ)+ < κ ≤ λ, TFAEV – Every DSD space X with |X| = ∆(X) = λ is κ-resolvable. – Every MN space X with |X| = ∆(X) = λ is κ-resolvable. – For every λ-filtration F, the space X(F) is κ-resolvable.

NOTE. For maximal resolvability, the cases κ = λ are of interest.

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the two steps of reduction

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the two steps of reduction

Lemma 1. [J-S-Sz]

If λ is regular, X is DSD with |X| = ∆(X) = λ, and there are "dense many" points in X that are not CAPs of any SD set of size λ, then X is λ-resolvable.

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the two steps of reduction

Lemma 1. [J-S-Sz]

If λ is regular, X is DSD with |X| = ∆(X) = λ, and there are "dense many" points in X that are not CAPs of any SD set of size λ, then X is λ-resolvable.

Lemma 2. [J-S-Sz]

For any λ ≥ ω, if X is any space s.t. every point in X is the CAP of some SD set of size λ, then there is a full λ-filtration F and a one-one continuous map g : X(F) → X .

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the two steps of reduction

Lemma 1. [J-S-Sz]

If λ is regular, X is DSD with |X| = ∆(X) = λ, and there are "dense many" points in X that are not CAPs of any SD set of size λ, then X is λ-resolvable.

Lemma 2. [J-S-Sz]

For any λ ≥ ω, if X is any space s.t. every point in X is the CAP of some SD set of size λ, then there is a full λ-filtration F and a one-one continuous map g : X(F) → X . This takes care of the case when λ is regular.

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the two steps of reduction

Lemma 1. [J-S-Sz]

If λ is regular, X is DSD with |X| = ∆(X) = λ, and there are "dense many" points in X that are not CAPs of any SD set of size λ, then X is λ-resolvable.

Lemma 2. [J-S-Sz]

For any λ ≥ ω, if X is any space s.t. every point in X is the CAP of some SD set of size λ, then there is a full λ-filtration F and a one-one continuous map g : X(F) → X . This takes care of the case when λ is regular. The singular case (proved in [J-M]) is similar but more complicated.

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λ-resolvability of λ-filtration spaces

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λ-resolvability of λ-filtration spaces

THEOREM [J-M]

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λ-resolvability of λ-filtration spaces

THEOREM [J-M]

If κ ≤ λ and F is a λ-filtration s.t.

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λ-resolvability of λ-filtration spaces

THEOREM [J-M]

If κ ≤ λ and F is a λ-filtration s.t. (i) for every t ∈ T = dom(F), if µt ≥ κ then F(t) is κ-decomposable,

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λ-resolvability of λ-filtration spaces

THEOREM [J-M]

If κ ≤ λ and F is a λ-filtration s.t. (i) for every t ∈ T = dom(F), if µt ≥ κ then F(t) is κ-decomposable, (ii) for every t ∈ T = dom(F) and µ ≤ κ, {α < µt : F(tα) is µ-decomposable} ∈ F(t) ,

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λ-resolvability of λ-filtration spaces

THEOREM [J-M]

If κ ≤ λ and F is a λ-filtration s.t. (i) for every t ∈ T = dom(F), if µt ≥ κ then F(t) is κ-decomposable, (ii) for every t ∈ T = dom(F) and µ ≤ κ, {α < µt : F(tα) is µ-decomposable} ∈ F(t) , then X(F) is κ-resolvable.

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λ-resolvability of λ-filtration spaces

THEOREM [J-M]

If κ ≤ λ and F is a λ-filtration s.t. (i) for every t ∈ T = dom(F), if µt ≥ κ then F(t) is κ-decomposable, (ii) for every t ∈ T = dom(F) and µ ≤ κ, {α < µt : F(tα) is µ-decomposable} ∈ F(t) , then X(F) is κ-resolvable.

COROLLARY [J-M]

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λ-resolvability of λ-filtration spaces

THEOREM [J-M]

If κ ≤ λ and F is a λ-filtration s.t. (i) for every t ∈ T = dom(F), if µt ≥ κ then F(t) is κ-decomposable, (ii) for every t ∈ T = dom(F) and µ ≤ κ, {α < µt : F(tα) is µ-decomposable} ∈ F(t) , then X(F) is κ-resolvable.

COROLLARY [J-M]

If every F ∈ un(µ) is maximally decomposable whenever ω ≤ µ ≤ λ, then X(F) is λ-resolvable for any λ-filtration F.

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