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Resolvability properties of certain topological spaces Istvn Juhsz Alfrd Rnyi Institute of Mathematics Sao Paulo, Brasil, August 2013 Istvn Juhsz (Rnyi Institute) Resolvability Sao Paulo 2013 1 / 18 resolvability Istvn

  1. Malychin’s problem EXAMPLE. (Hewitt, ’43) There is a countable T 3 space X that is – crowded (i.e. ∆( X ) = | X | = ℵ 0 ) and – irresolvable( ≡ not 2-resolvable). PROBLEM. (Malychin, 1995) Is a Lindelöf T 3 space X with ∆( X ) > ω resolvable? NOTE. Malychin constructed Lindelöf irresolvable Hausdorff (= T 2 ) spaces, and Pavlov Lindelöf irresolvable Uryson (= T 2 . 5 ) spaces. István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 4 / 18

  2. Malychin’s problem EXAMPLE. (Hewitt, ’43) There is a countable T 3 space X that is – crowded (i.e. ∆( X ) = | X | = ℵ 0 ) and – irresolvable( ≡ not 2-resolvable). PROBLEM. (Malychin, 1995) Is a Lindelöf T 3 space X with ∆( X ) > ω resolvable? NOTE. Malychin constructed Lindelöf irresolvable Hausdorff (= T 2 ) spaces, and Pavlov Lindelöf irresolvable Uryson (= T 2 . 5 ) spaces. THEOREM. (Filatova, 2004) YES, every Lindelöf T 3 space X with ∆( X ) > ω is 2-resolvable. István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 4 / 18

  3. Malychin’s problem EXAMPLE. (Hewitt, ’43) There is a countable T 3 space X that is – crowded (i.e. ∆( X ) = | X | = ℵ 0 ) and – irresolvable( ≡ not 2-resolvable). PROBLEM. (Malychin, 1995) Is a Lindelöf T 3 space X with ∆( X ) > ω resolvable? NOTE. Malychin constructed Lindelöf irresolvable Hausdorff (= T 2 ) spaces, and Pavlov Lindelöf irresolvable Uryson (= T 2 . 5 ) spaces. THEOREM. (Filatova, 2004) YES, every Lindelöf T 3 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

  4. Malychin’s problem EXAMPLE. (Hewitt, ’43) There is a countable T 3 space X that is – crowded (i.e. ∆( X ) = | X | = ℵ 0 ) and – irresolvable( ≡ not 2-resolvable). PROBLEM. (Malychin, 1995) Is a Lindelöf T 3 space X with ∆( X ) > ω resolvable? NOTE. Malychin constructed Lindelöf irresolvable Hausdorff (= T 2 ) spaces, and Pavlov Lindelöf irresolvable Uryson (= T 2 . 5 ) spaces. THEOREM. (Filatova, 2004) YES, every Lindelöf T 3 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

  5. Pavlov’s theorems István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 5 / 18

  6. Pavlov’s theorems sup {| D | : D ⊂ X is discrete } s ( X ) = István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 5 / 18

  7. Pavlov’s theorems sup {| D | : D ⊂ X is discrete } s ( X ) = sup {| D | : D ⊂ X is closed discrete } e ( X ) = István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 5 / 18

  8. Pavlov’s theorems sup {| D | : D ⊂ X is discrete } s ( X ) = sup {| D | : D ⊂ X is closed discrete } e ( X ) = THEOREM. (Pavlov, 2002) István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 5 / 18

  9. Pavlov’s theorems sup {| D | : D ⊂ X is discrete } s ( X ) = sup {| D | : D ⊂ X is closed discrete } e ( X ) = THEOREM. (Pavlov, 2002) (i) Any T 2 space X with ∆( X ) > s ( X ) + is maximally resolvable. István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 5 / 18

  10. Pavlov’s theorems sup {| D | : D ⊂ X is discrete } s ( X ) = sup {| D | : D ⊂ X is closed discrete } e ( X ) = THEOREM. (Pavlov, 2002) (i) Any T 2 space X with ∆( X ) > s ( X ) + is maximally resolvable. (ii) Any T 3 space X with ∆( X ) > e ( X ) + is ω -resolvable. István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 5 / 18

  11. Pavlov’s theorems sup {| D | : D ⊂ X is discrete } s ( X ) = sup {| D | : D ⊂ X is closed discrete } e ( X ) = THEOREM. (Pavlov, 2002) (i) Any T 2 space X with ∆( X ) > s ( X ) + is maximally resolvable. (ii) Any T 3 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

  12. Pavlov’s theorems sup {| D | : D ⊂ X is discrete } s ( X ) = sup {| D | : D ⊂ X is closed discrete } e ( X ) = THEOREM. (Pavlov, 2002) (i) Any T 2 space X with ∆( X ) > s ( X ) + is maximally resolvable. (ii) Any T 3 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 T 3 space X with ∆( X ) > e ( X ) is ω -resolvable. István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 5 / 18

  13. Pavlov’s theorems sup {| D | : D ⊂ X is discrete } s ( X ) = sup {| D | : D ⊂ X is closed discrete } e ( X ) = THEOREM. (Pavlov, 2002) (i) Any T 2 space X with ∆( X ) > s ( X ) + is maximally resolvable. (ii) Any T 3 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 T 3 space X with ∆( X ) > e ( X ) is ω -resolvable. In particular, every Lindelöf T 3 space X with ∆( X ) > ω is ω -resolvable. István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 5 / 18

  14. Pavlov’s theorems sup {| D | : D ⊂ X is discrete } s ( X ) = sup {| D | : D ⊂ X is closed discrete } e ( X ) = THEOREM. (Pavlov, 2002) (i) Any T 2 space X with ∆( X ) > s ( X ) + is maximally resolvable. (ii) Any T 3 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 T 3 space X with ∆( X ) > e ( X ) is ω -resolvable. In particular, every Lindelöf T 3 space X with ∆( X ) > ω is ω -resolvable. István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 5 / 18

  15. J-S-Sz István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 6 / 18

  16. 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

  17. 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 T 3 , ∆( X ) ≥ κ = cf ( κ ) > ω and X has no closed discrete subset of size κ then X is ω -resolvable. István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 6 / 18

  18. 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 T 3 , ∆( X ) ≥ κ = cf ( κ ) > ω and X has no closed discrete subset of size κ then X is ω -resolvable. NOTE. For ∆( X ) > ω regular these suffice. István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 6 / 18

  19. 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 T 3 , ∆( X ) ≥ κ = cf ( κ ) > ω and X has no closed discrete subset of 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

  20. 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 T 3 , ∆( X ) ≥ κ = cf ( κ ) > ω and X has no closed discrete subset of 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

  21. 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 T 3 , ∆( X ) ≥ κ = cf ( κ ) > ω and X has no closed discrete subset of 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

  22. 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 T 3 , ∆( X ) ≥ κ = cf ( κ ) > ω and X has no closed discrete subset of 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

  23. < λ -resolvable István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 7 / 18

  24. < λ -resolvable THEOREM. (J-S-Sz, 2006) For any κ ≥ λ = cf ( λ ) > ω there is a dense X ⊂ D ( 2 ) 2 κ with ∆( X ) = κ that is < λ -resolvable but not λ -resolvable. István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 7 / 18

  25. < λ -resolvable THEOREM. (J-S-Sz, 2006) For any κ ≥ λ = cf ( λ ) > ω there is a dense X ⊂ D ( 2 ) 2 κ 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

  26. < λ -resolvable THEOREM. (J-S-Sz, 2006) For any κ ≥ λ = cf ( λ ) > ω there is a dense X ⊂ D ( 2 ) 2 κ 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

  27. < λ -resolvable THEOREM. (J-S-Sz, 2006) For any κ ≥ λ = cf ( λ ) > ω there is a dense X ⊂ D ( 2 ) 2 κ 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

  28. < λ -resolvable THEOREM. (J-S-Sz, 2006) For any κ ≥ λ = cf ( λ ) > ω there is a dense X ⊂ D ( 2 ) 2 κ 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

  29. < λ -resolvable THEOREM. (J-S-Sz, 2006) For any κ ≥ λ = cf ( λ ) > ω there is a dense X ⊂ D ( 2 ) 2 κ 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 = ℵ ω 1 ? István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 7 / 18

  30. < λ -resolvable THEOREM. (J-S-Sz, 2006) For any κ ≥ λ = cf ( λ ) > ω there is a dense X ⊂ D ( 2 ) 2 κ 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 = ℵ ω 1 ? István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 7 / 18

  31. monotone normality István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 8 / 18

  32. monotone normality DEFINITION. István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 8 / 18

  33. monotone normality DEFINITION. The space X is monotonically normal ( MN ) iff it is T 1 (i.e. all singletons are closed) István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 8 / 18

  34. monotone normality DEFINITION. The space X is monotonically normal ( MN ) iff it is T 1 (i.e. all singletons are closed) and it has a monotone normality operator H that "halves" neighbourhoods : István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 8 / 18

  35. monotone normality DEFINITION. The space X is monotonically normal ( MN ) iff it is T 1 (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. István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 8 / 18

  36. monotone normality DEFINITION. The space X is monotonically normal ( MN ) iff it is T 1 (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 , István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 8 / 18

  37. monotone normality DEFINITION. The space X is monotonically normal ( MN ) iff it is T 1 (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 . István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 8 / 18

  38. monotone normality DEFINITION. The space X is monotonically normal ( MN ) iff it is T 1 (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 István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 8 / 18

  39. monotone normality DEFINITION. The space X is monotonically normal ( MN ) iff it is T 1 (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. István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 8 / 18

  40. monotone normality DEFINITION. The space X is monotonically normal ( MN ) iff it is T 1 (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? István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 8 / 18

  41. SD spaces István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 9 / 18

  42. SD spaces DEFINITION. István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 9 / 18

  43. SD spaces DEFINITION. (i) D ⊂ X is strongly discrete if there are pairwise disjoint open sets { U x : x ∈ D } with x ∈ U x for x ∈ D . István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 9 / 18

  44. SD spaces DEFINITION. (i) D ⊂ X is strongly discrete if there are pairwise disjoint open sets { U x : x ∈ D } with x ∈ U x for x ∈ D . EXAMPLE: Countable discrete sets in T 3 spaces are SD. István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 9 / 18

  45. SD spaces DEFINITION. (i) D ⊂ X is strongly discrete if there are pairwise disjoint open sets { U x : x ∈ D } with x ∈ U x for x ∈ D . EXAMPLE: Countable discrete sets in T 3 spaces are SD. (ii) X is an SD space if every non-isolated point x ∈ X is an SD limit. István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 9 / 18

  46. SD spaces DEFINITION. (i) D ⊂ X is strongly discrete if there are pairwise disjoint open sets { U x : x ∈ D } with x ∈ U x for x ∈ D . EXAMPLE: Countable discrete sets in T 3 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 T 1 crowded SD space is ω -resolvable. István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 9 / 18

  47. SD spaces DEFINITION. (i) D ⊂ X is strongly discrete if there are pairwise disjoint open sets { U x : x ∈ D } with x ∈ U x for x ∈ D . EXAMPLE: Countable discrete sets in T 3 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 T 1 crowded SD space is ω -resolvable. THEOREM. (DTTW, 2002) MN spaces are SD, hence crowded MN spaces are ω -resolvable. István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 9 / 18

  48. SD spaces DEFINITION. (i) D ⊂ X is strongly discrete if there are pairwise disjoint open sets { U x : x ∈ D } with x ∈ U x for x ∈ D . EXAMPLE: Countable discrete sets in T 3 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 T 1 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? István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 9 / 18

  49. [J-S-Sz] István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 10 / 18

  50. [J-S-Sz] [J-S-Sz] ≡ I. J UHÁSZ , L. S OUKUP AND Z. S ZENTMIKLÓSSY , Resolvability and monotone normality , Israel J. Math., 166 (2008), no. 1, pp. 1–16. István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 10 / 18

  51. [J-S-Sz] [J-S-Sz] ≡ I. J UHÁSZ , L. S OUKUP AND Z. S ZENTMIKLÓSSY , 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. István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 10 / 18

  52. [J-S-Sz] [J-S-Sz] ≡ I. J UHÁSZ , L. S OUKUP AND Z. S ZENTMIKLÓSSY , 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. István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 10 / 18

  53. [J-S-Sz] [J-S-Sz] ≡ I. J UHÁSZ , L. S OUKUP AND Z. S ZENTMIKLÓSSY , 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] István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 10 / 18

  54. [J-S-Sz] [J-S-Sz] ≡ I. J UHÁSZ , L. S OUKUP AND Z. S ZENTMIKLÓSSY , 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 ω 1 -irrresolvable. István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 10 / 18

  55. [J-S-Sz] [J-S-Sz] ≡ I. J UHÁSZ , L. S OUKUP AND Z. S ZENTMIKLÓSSY , 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 ω 1 -irrresolvable. – If X is DSD with | X | < ℵ ω then X is maximally resolvable. István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 10 / 18

  56. [J-S-Sz] [J-S-Sz] ≡ I. J UHÁSZ , L. S OUKUP AND Z. S ZENTMIKLÓSSY , 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 ω 1 -irrresolvable. – If X is DSD with | X | < ℵ ω then X is maximally resolvable. – From a supercompact cardinal, it is consistent to have a MN space X with | X | = ∆( X ) = ℵ ω that is ω 2 -irresolvable. István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 10 / 18

  57. [J-S-Sz] [J-S-Sz] ≡ I. J UHÁSZ , L. S OUKUP AND Z. S ZENTMIKLÓSSY , 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 ω 1 -irrresolvable. – If X is DSD with | X | < ℵ ω then X is maximally resolvable. – From a supercompact cardinal, it is consistent to have a MN space X with | X | = ∆( X ) = ℵ ω that is ω 2 -irresolvable. This left a number of questions open. István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 10 / 18

  58. decomposability of ultrafilters István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 11 / 18

  59. decomposability of ultrafilters DEFINITION. István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 11 / 18

  60. decomposability of ultrafilters DEFINITION. – An ultrafilter F is µ -descendingly complete iff for any descending µ -sequence { A α : α < µ } ⊂ F we have � { A α : α < µ } ∈ F . István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 11 / 18

  61. decomposability of ultrafilters DEFINITION. – An ultrafilter F is µ -descendingly complete iff for any descending µ -sequence { A α : α < µ } ⊂ F we have � { A α : α < µ } ∈ F . µ -descendingly incomplete is (now) called µ -decomposable. István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 11 / 18

  62. decomposability of ultrafilters DEFINITION. – An ultrafilter F is µ -descendingly complete iff for any descending µ -sequence { A α : α < µ } ⊂ F we have � { A α : α < µ } ∈ F . µ -descendingly incomplete is (now) called µ -decomposable. – ∆( F ) = min {| A | : A ∈ F} . István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 11 / 18

  63. decomposability of ultrafilters DEFINITION. – An ultrafilter F is µ -descendingly complete iff for any descending µ -sequence { A α : α < µ } ⊂ F we have � { A α : 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 ) . István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 11 / 18

  64. decomposability of ultrafilters DEFINITION. – An ultrafilter F is µ -descendingly complete iff for any descending µ -sequence { A α : α < µ } ⊂ F we have � { A α : 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. István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 11 / 18

  65. decomposability of ultrafilters DEFINITION. – An ultrafilter F is µ -descendingly complete iff for any descending µ -sequence { A α : α < µ } ⊂ F we have � { A α : 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. István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 11 / 18

  66. decomposability of ultrafilters DEFINITION. – An ultrafilter F is µ -descendingly complete iff for any descending µ -sequence { A α : α < µ } ⊂ F we have � { A α : 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. István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 11 / 18

  67. decomposability of ultrafilters DEFINITION. – An ultrafilter F is µ -descendingly complete iff for any descending µ -sequence { A α : α < µ } ⊂ F we have � { A α : 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 ) < ℵ ω is maximally decomposable. István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 11 / 18

  68. [J-M] István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 12 / 18

  69. [J-M] [J-M] ≡ I. J UHÁSZ AND M. M AGIDOR , On the maximal resolvability of monotonically normal spaces , Israel J. Math, 192 (2012), 637-666. István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 12 / 18

  70. [J-M] [J-M] ≡ I. J UHÁSZ AND M. M AGIDOR , On the maximal resolvability of monotonically normal spaces , Israel J. Math, 192 (2012), 637-666. Main results of [J-M] István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 12 / 18

  71. [J-M] [J-M] ≡ I. J UHÁSZ AND M. M AGIDOR , On the maximal resolvability of monotonically normal spaces , Israel J. Math, 192 (2012), 637-666. Main results of [J-M] (1) TFAEV István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 12 / 18

  72. [J-M] [J-M] ≡ I. J UHÁSZ AND M. M AGIDOR , 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. István Juhász (Rényi Institute) Resolvability Sao Paulo 2013 12 / 18

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