resolvability properties of certain topological spaces
play

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

Mar 16, 2023 •235 likes •1.78k views

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

Recommend

Equilogical spaces A Chu-like Extension of Topological Spaces There are several constructions

Equilogical spaces A Chu-like Extension of Topological Spaces There are several constructions

Equilogical spaces A Chu-like Extension of Topological Spaces There are several constructions that embed the traditional category of topological spaces and continuous functions in a cartesian closed category. Paul Taylor (All topological

119 views • 7 slides
Topological properties of convolutor spaces via the short-time Fourier transform Andreas

Topological properties of convolutor spaces via the short-time Fourier transform Andreas

Topological properties of convolutor spaces via the short-time Fourier transform Andreas Debrouwere (Joint work with Jasson Vindas) Ghent University Pawel Doma nski Memorial Conference 6 July 2018 1 / 14 The space of integrable

582 views • 54 slides
Explicit Construction of Multiple Access Channel Resolvability Codes from Source Resolvability

Explicit Construction of Multiple Access Channel Resolvability Codes from Source Resolvability

Explicit Construction of Multiple Access Channel Resolvability Codes from Source Resolvability Codes Rumia Sultana R emi A. Chou Wichita State University 2020 IEEE International Symposium on Information Theory 1/29 Outline Introduction

296 views • 29 slides
Topological algebras on Boolean spaces as dual spaces and applications in formal language theory

Topological algebras on Boolean spaces as dual spaces and applications in formal language theory

Topological algebras on Boolean spaces as dual spaces and applications in formal language theory Mai Gehrke CNRS and Paris Diderot Stone duality DL 0-dimCompHaus BA dual equivalence Spectral spaces Stably compact spaces Proximity

702 views • 33 slides
On the reversibility of topological spaces and related notions. Vitalij A.Chatyrko May 25, 2018

On the reversibility of topological spaces and related notions. Vitalij A.Chatyrko May 25, 2018

On the reversibility of topological spaces and related notions. Vitalij A.Chatyrko May 25, 2018 1 / 26 Content Introduction. (Non)reversible topological spaces. The reversibility of separable metric connected manifolds. A generalization of

897 views • 26 slides
Topoi generated by topological spaces A generalization of Johnstones topos Topoi

Topoi generated by topological spaces A generalization of Johnstones topos Topoi

Topoi generated by topological spaces A generalization of Johnstones topos Topoi generated by topological monoids Jos e Reinaldo Monta nez Puentes jrmontanezp@unal.edu.co Departamento de Matem aticas Universidad Nacional

162 views • 12 slides
Topological aspects of perverse sheaves Jon Woolf June, 2017 Part I Stratified spaces

Topological aspects of perverse sheaves Jon Woolf June, 2017 Part I Stratified spaces

Topological aspects of perverse sheaves Jon Woolf June, 2017 Part I Stratified spaces Stratifications A stratification of a topological space X consists of 1. a decomposition X = i S i into disjoint locally-closed subspaces 2. geometric

607 views • 41 slides
Topological spaces of monadic MV-algebras R. Grigolia (Tbilisi), A. Di Nola and G. Lenzi (Salerno)

Topological spaces of monadic MV-algebras R. Grigolia (Tbilisi), A. Di Nola and G. Lenzi (Salerno)

Topological spaces of monadic MV-algebras R. Grigolia (Tbilisi), A. Di Nola and G. Lenzi (Salerno) TACL 2017 Prague, 26-30 June 2017 Topological spaces of monadic MV-algebras G. Lenzi Summary We generalize Belluce functor from MV-algebras

1.24k views • 122 slides
Topological properties in solids probed by Experiment - Quantum Hall effect - 2D TIs - Weak

Topological properties in solids probed by Experiment - Quantum Hall effect - 2D TIs - Weak

Markus Morgenstern Topological properties in solids probed by Experiment - Quantum Hall effect - 2D TIs - Weak Topological Insulators - Strong topological insulators - Vienna, 04.08.2014 What is Topology ? Wikipedia: Topology is the

1.21k views • 33 slides
Topological Insulator Literal meaning of Topology Properties that are preserved under

Topological Insulator Literal meaning of Topology Properties that are preserved under

Topological Insulator Literal meaning of Topology Properties that are preserved under continuous deformation. No tearing, no gluing (Mathematical) Landau symmetry-breaking theory Same atoms but different properties Landau

661 views • 16 slides
Some topological properties a b of planar lambda terms f g Noam Zeilberger h k

Some topological properties a b of planar lambda terms f g Noam Zeilberger h k

Some topological properties a b of planar lambda terms f g Noam Zeilberger h k (work-in-progress with Jason Reed) e c i CLA 2019 j Versailles, 1-2 July d [Background] A few views on maps T opological de fi nition map = 2-cell

872 views • 50 slides
On some topological properties of pointfree function rings Mark Sioen BLAST 2013, Chapman

On some topological properties of pointfree function rings Mark Sioen BLAST 2013, Chapman

On some topological properties of pointfree function rings Mark Sioen BLAST 2013, Chapman University, August 59 Introduction Spatial setting Frames and L ( R ) Topologies on R ( L ) and R ( L ) Dini properties The S-W property

1.67k views • 122 slides
Supports and approximation properties in Lipschitz-free spaces Eva Perneck a Czech Technical

Supports and approximation properties in Lipschitz-free spaces Eva Perneck a Czech Technical

Lipschitz-free spaces Supports Approximation properties Supports and approximation properties in Lipschitz-free spaces Eva Perneck a Czech Technical University, Prague Workshop on Banach spaces and Banach lattices Madrid, September 2019

1.14k views • 83 slides
G -bases in free objects of Topological Algebra (Local) -bases in topological and uniform

G -bases in free objects of Topological Algebra (Local) -bases in topological and uniform

G -bases in free objects of Topological Algebra (Local) -bases in topological and uniform spaces Taras Banakh and Arkady Leiderman Lviv &amp; Kielce Prague, 29 July 2016 T.Banakh G -bases in free objects of Topological Algebra (Local)

1.48k views • 102 slides
Fold maps, topological information of their Reeb spaces and their source manifolds. (

Fold maps, topological information of their Reeb spaces and their source manifolds. (

Fold maps, topological information of their Reeb spaces and their source manifolds. ( Reeb ) Naoki Kitazawa Institute of Mathematics for Industry (IMI),

632 views • 32 slides
1 Topological Properties (17 March) a face), or has an ( n 1) -dimensional face with three or

1 Topological Properties (17 March) a face), or has an ( n 1) -dimensional face with three or

1 Topological Properties (17 March) a face), or has an ( n 1) -dimensional face with three or more n -dimensional simplices adjacent to it, then this space is not a manifold. For example, the following topo- The previous two lectures we

147 views • 3 slides
Topological essen-als Get the basics right and the rest

Topological essen-als Get the basics right and the rest

Topological essen-als Get the basics right and the rest will follow (Barry Ninham, founding professor of Applied Maths, RSPE) 1. Spaces,

816 views • 26 slides
Ramsey properties and ultrahomogeneity for Banach and operator spaces. J. Lopez-Abad Instituto

Ramsey properties and ultrahomogeneity for Banach and operator spaces. J. Lopez-Abad Instituto

Ramsey properties and ultrahomogeneity for Banach and operator spaces. J. Lopez-Abad Instituto de Ciencias Matem aticas,CSIC, Madrid joint work with D. Barto sov a, M. Lupini and B. Mbombo Transfinite methods in Banach spaces and

687 views • 66 slides
Approximation properties in Lipschitz-free spaces over groups Pedro L. Kaufmann Federal

Approximation properties in Lipschitz-free spaces over groups Pedro L. Kaufmann Federal

Approximation properties in Lipschitz-free spaces over groups Pedro L. Kaufmann Federal University of S ao Paulo ICMAT Workshop on Banach Spaces and Banach Lattices September 11th, 2019 Supported by FAPESP grants 2017/18623-5,

521 views • 25 slides
Homogeneous spaces as coset spaces of groups from special classes K. Kozlov Lomonosov Moscow

Homogeneous spaces as coset spaces of groups from special classes K. Kozlov Lomonosov Moscow

Example I Partial answer on Questions 1, 2 Example II Decompositions of actions Homogeneous spaces as coset spaces of groups from special classes K. Kozlov Lomonosov Moscow State University PRAGUE TOPOLOGICAL SYMPOSIUM July 2016 K. Kozlov

843 views • 83 slides
The Fundamental Group of Topological Spaces An Introduction To Algebraic Topology Thomas Gagne

The Fundamental Group of Topological Spaces An Introduction To Algebraic Topology Thomas Gagne

Basic Topology The Fundamental Group Examples Seifert-van Kampen Theorem The Fundamental Group of Topological Spaces An Introduction To Algebraic Topology Thomas Gagne Department of Mathematics University of Puget Sound May 4, 2015 Basic

393 views • 22 slides
Topological Properties of the Air Navigation Route System using Complex Network Theory Xiaoqian

Topological Properties of the Air Navigation Route System using Complex Network Theory Xiaoqian

www.DLR.de Chart 1 Topological Properties of the Air Navigation Route System using Complex Network Theory Xiaoqian Sun 1 , Sebastian Wandelt 2 , and Florian Linke 1 1 Institute of Air Transportation Systems, German Aerospace Center 2

415 views • 22 slides
I -continuity in topological spaces 1. Definitions Definition 1.1. A family I of subsets of N is

I -continuity in topological spaces 1. Definitions Definition 1.1. A family I of subsets of N is

I -continuity in topological spaces 1. Definitions Definition 1.1. A family I of subsets of N is an ideal in N if (1) A, B I A B I , (2) A I and B A B I . Let us call an ideal I in N proper if N / I . I is

78 views • 5 slides
Enriched Topologies and Topological Representation of Semi-Unital Quantales Ulrich H ohle

Enriched Topologies and Topological Representation of Semi-Unital Quantales Ulrich H ohle

Enriched Topologies and Topological Representation of Semi-Unital Quantales Ulrich H ohle Bergische Universit at, Wuppertal, Germany Coimbra, September 2018 Table of Contents 1 Terminology and Motivation 2 Enriched Topological Spaces 3

1.1k views • 62 slides

More recommend