Between Polish and completely Baire
Abstract All spaces are assumed to be separable and metrizable. Consider the following properties of a space X. X is Polish.For every countable crowded Q⊆X%${Q \subseteq X}%$ there exists a crowded Q′⊆Q%${Q'\subseteq Q}%$ with compact closure.Every closed subspace of X is either scattered or it...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Medini, Andrea [verfasserIn] |
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| Format: |
E-Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2014 |
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| Schlagwörter: |
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| Anmerkung: |
© Springer-Verlag Berlin Heidelberg 2014 |
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| Übergeordnetes Werk: |
Enthalten in: Archive for mathematical logic - Berlin : Springer, 1950, 54(2014), 1-2 vom: 30. Okt., Seite 231-245 |
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| Übergeordnetes Werk: |
volume:54 ; year:2014 ; number:1-2 ; day:30 ; month:10 ; pages:231-245 |
| Links: |
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| DOI / URN: |
10.1007/s00153-014-0409-4 |
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| Katalog-ID: |
SPR001300679 |
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| 520 | |a Abstract All spaces are assumed to be separable and metrizable. Consider the following properties of a space X. X is Polish.For every countable crowded Q⊆X%${Q \subseteq X}%$ there exists a crowded Q′⊆Q%${Q'\subseteq Q}%$ with compact closure.Every closed subspace of X is either scattered or it contains a homeomorphic copy of 2ω%${2^\omega}%$.Every closed subspace of X is a Baire space. While (4) is the well-known property of being completely Baire, properties (2) and (3) have been recently introduced by Kunen, Medini and Zdomskyy, who named them the Miller property and the Cantor-Bendixson property respectively. It turns out that the implications %${(1) \rightarrow (2) \rightarrow (3) \rightarrow (4)}%$ hold for every space X. Furthermore, it follows from a classical result of Hurewicz that all these implications are equivalences if X is coanalytic. Under the axiom of Projective Determinacy, this equivalence result extends to all projective spaces. We will complete the picture by giving a %${\sf ZFC}%$ counterexample and a consistent definable counterexample of lowest possible complexity to the implication %${(i) \leftarrow (i + 1)}%$ for %${i = 1, 2, 3}%$. For one of these counterexamples we will need a classical theorem of Martin and Solovay, of which we give a new proof, based on a result of Baldwin and Beaudoin. Finally, using a method of Fischer and Friedman, we will investigate how changing the value of the continuum affects the definability of these counterexamples. Along the way, we will show that every uncountable completely Baire space has size continuum. | ||
| 650 | 4 | |a Polish |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Miller property |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Cantor-Bendixson property |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Completely Baire |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Hereditarily Baire |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Miller-measurable |7 (dpeaa)DE-He213 | |
| 700 | 1 | |a Zdomskyy, Lyubomyr |4 aut | |
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10.1007/s00153-014-0409-4 doi (DE-627)SPR001300679 (SPR)s00153-014-0409-4-e DE-627 ger DE-627 rakwb eng Medini, Andrea verfasserin aut Between Polish and completely Baire 2014 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer-Verlag Berlin Heidelberg 2014 Abstract All spaces are assumed to be separable and metrizable. Consider the following properties of a space X. X is Polish.For every countable crowded Q⊆X%${Q \subseteq X}%$ there exists a crowded Q′⊆Q%${Q'\subseteq Q}%$ with compact closure.Every closed subspace of X is either scattered or it contains a homeomorphic copy of 2ω%${2^\omega}%$.Every closed subspace of X is a Baire space. While (4) is the well-known property of being completely Baire, properties (2) and (3) have been recently introduced by Kunen, Medini and Zdomskyy, who named them the Miller property and the Cantor-Bendixson property respectively. It turns out that the implications %${(1) \rightarrow (2) \rightarrow (3) \rightarrow (4)}%$ hold for every space X. Furthermore, it follows from a classical result of Hurewicz that all these implications are equivalences if X is coanalytic. Under the axiom of Projective Determinacy, this equivalence result extends to all projective spaces. We will complete the picture by giving a %${\sf ZFC}%$ counterexample and a consistent definable counterexample of lowest possible complexity to the implication %${(i) \leftarrow (i + 1)}%$ for %${i = 1, 2, 3}%$. For one of these counterexamples we will need a classical theorem of Martin and Solovay, of which we give a new proof, based on a result of Baldwin and Beaudoin. Finally, using a method of Fischer and Friedman, we will investigate how changing the value of the continuum affects the definability of these counterexamples. Along the way, we will show that every uncountable completely Baire space has size continuum. Polish (dpeaa)DE-He213 Miller property (dpeaa)DE-He213 Cantor-Bendixson property (dpeaa)DE-He213 Completely Baire (dpeaa)DE-He213 Hereditarily Baire (dpeaa)DE-He213 Miller-measurable (dpeaa)DE-He213 Zdomskyy, Lyubomyr aut Enthalten in Archive for mathematical logic Berlin : Springer, 1950 54(2014), 1-2 vom: 30. Okt., Seite 231-245 (DE-627)235503215 (DE-600)1398309-X 1432-0665 nnns volume:54 year:2014 number:1-2 day:30 month:10 pages:231-245 https://dx.doi.org/10.1007/s00153-014-0409-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_206 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 54 2014 1-2 30 10 231-245 |
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10.1007/s00153-014-0409-4 doi (DE-627)SPR001300679 (SPR)s00153-014-0409-4-e DE-627 ger DE-627 rakwb eng Medini, Andrea verfasserin aut Between Polish and completely Baire 2014 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer-Verlag Berlin Heidelberg 2014 Abstract All spaces are assumed to be separable and metrizable. Consider the following properties of a space X. X is Polish.For every countable crowded Q⊆X%${Q \subseteq X}%$ there exists a crowded Q′⊆Q%${Q'\subseteq Q}%$ with compact closure.Every closed subspace of X is either scattered or it contains a homeomorphic copy of 2ω%${2^\omega}%$.Every closed subspace of X is a Baire space. While (4) is the well-known property of being completely Baire, properties (2) and (3) have been recently introduced by Kunen, Medini and Zdomskyy, who named them the Miller property and the Cantor-Bendixson property respectively. It turns out that the implications %${(1) \rightarrow (2) \rightarrow (3) \rightarrow (4)}%$ hold for every space X. Furthermore, it follows from a classical result of Hurewicz that all these implications are equivalences if X is coanalytic. Under the axiom of Projective Determinacy, this equivalence result extends to all projective spaces. We will complete the picture by giving a %${\sf ZFC}%$ counterexample and a consistent definable counterexample of lowest possible complexity to the implication %${(i) \leftarrow (i + 1)}%$ for %${i = 1, 2, 3}%$. For one of these counterexamples we will need a classical theorem of Martin and Solovay, of which we give a new proof, based on a result of Baldwin and Beaudoin. Finally, using a method of Fischer and Friedman, we will investigate how changing the value of the continuum affects the definability of these counterexamples. Along the way, we will show that every uncountable completely Baire space has size continuum. Polish (dpeaa)DE-He213 Miller property (dpeaa)DE-He213 Cantor-Bendixson property (dpeaa)DE-He213 Completely Baire (dpeaa)DE-He213 Hereditarily Baire (dpeaa)DE-He213 Miller-measurable (dpeaa)DE-He213 Zdomskyy, Lyubomyr aut Enthalten in Archive for mathematical logic Berlin : Springer, 1950 54(2014), 1-2 vom: 30. Okt., Seite 231-245 (DE-627)235503215 (DE-600)1398309-X 1432-0665 nnns volume:54 year:2014 number:1-2 day:30 month:10 pages:231-245 https://dx.doi.org/10.1007/s00153-014-0409-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_206 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 54 2014 1-2 30 10 231-245 |
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10.1007/s00153-014-0409-4 doi (DE-627)SPR001300679 (SPR)s00153-014-0409-4-e DE-627 ger DE-627 rakwb eng Medini, Andrea verfasserin aut Between Polish and completely Baire 2014 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer-Verlag Berlin Heidelberg 2014 Abstract All spaces are assumed to be separable and metrizable. Consider the following properties of a space X. X is Polish.For every countable crowded Q⊆X%${Q \subseteq X}%$ there exists a crowded Q′⊆Q%${Q'\subseteq Q}%$ with compact closure.Every closed subspace of X is either scattered or it contains a homeomorphic copy of 2ω%${2^\omega}%$.Every closed subspace of X is a Baire space. While (4) is the well-known property of being completely Baire, properties (2) and (3) have been recently introduced by Kunen, Medini and Zdomskyy, who named them the Miller property and the Cantor-Bendixson property respectively. It turns out that the implications %${(1) \rightarrow (2) \rightarrow (3) \rightarrow (4)}%$ hold for every space X. Furthermore, it follows from a classical result of Hurewicz that all these implications are equivalences if X is coanalytic. Under the axiom of Projective Determinacy, this equivalence result extends to all projective spaces. We will complete the picture by giving a %${\sf ZFC}%$ counterexample and a consistent definable counterexample of lowest possible complexity to the implication %${(i) \leftarrow (i + 1)}%$ for %${i = 1, 2, 3}%$. For one of these counterexamples we will need a classical theorem of Martin and Solovay, of which we give a new proof, based on a result of Baldwin and Beaudoin. Finally, using a method of Fischer and Friedman, we will investigate how changing the value of the continuum affects the definability of these counterexamples. Along the way, we will show that every uncountable completely Baire space has size continuum. Polish (dpeaa)DE-He213 Miller property (dpeaa)DE-He213 Cantor-Bendixson property (dpeaa)DE-He213 Completely Baire (dpeaa)DE-He213 Hereditarily Baire (dpeaa)DE-He213 Miller-measurable (dpeaa)DE-He213 Zdomskyy, Lyubomyr aut Enthalten in Archive for mathematical logic Berlin : Springer, 1950 54(2014), 1-2 vom: 30. Okt., Seite 231-245 (DE-627)235503215 (DE-600)1398309-X 1432-0665 nnns volume:54 year:2014 number:1-2 day:30 month:10 pages:231-245 https://dx.doi.org/10.1007/s00153-014-0409-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_206 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 54 2014 1-2 30 10 231-245 |
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10.1007/s00153-014-0409-4 doi (DE-627)SPR001300679 (SPR)s00153-014-0409-4-e DE-627 ger DE-627 rakwb eng Medini, Andrea verfasserin aut Between Polish and completely Baire 2014 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer-Verlag Berlin Heidelberg 2014 Abstract All spaces are assumed to be separable and metrizable. Consider the following properties of a space X. X is Polish.For every countable crowded Q⊆X%${Q \subseteq X}%$ there exists a crowded Q′⊆Q%${Q'\subseteq Q}%$ with compact closure.Every closed subspace of X is either scattered or it contains a homeomorphic copy of 2ω%${2^\omega}%$.Every closed subspace of X is a Baire space. While (4) is the well-known property of being completely Baire, properties (2) and (3) have been recently introduced by Kunen, Medini and Zdomskyy, who named them the Miller property and the Cantor-Bendixson property respectively. It turns out that the implications %${(1) \rightarrow (2) \rightarrow (3) \rightarrow (4)}%$ hold for every space X. Furthermore, it follows from a classical result of Hurewicz that all these implications are equivalences if X is coanalytic. Under the axiom of Projective Determinacy, this equivalence result extends to all projective spaces. We will complete the picture by giving a %${\sf ZFC}%$ counterexample and a consistent definable counterexample of lowest possible complexity to the implication %${(i) \leftarrow (i + 1)}%$ for %${i = 1, 2, 3}%$. For one of these counterexamples we will need a classical theorem of Martin and Solovay, of which we give a new proof, based on a result of Baldwin and Beaudoin. Finally, using a method of Fischer and Friedman, we will investigate how changing the value of the continuum affects the definability of these counterexamples. Along the way, we will show that every uncountable completely Baire space has size continuum. Polish (dpeaa)DE-He213 Miller property (dpeaa)DE-He213 Cantor-Bendixson property (dpeaa)DE-He213 Completely Baire (dpeaa)DE-He213 Hereditarily Baire (dpeaa)DE-He213 Miller-measurable (dpeaa)DE-He213 Zdomskyy, Lyubomyr aut Enthalten in Archive for mathematical logic Berlin : Springer, 1950 54(2014), 1-2 vom: 30. Okt., Seite 231-245 (DE-627)235503215 (DE-600)1398309-X 1432-0665 nnns volume:54 year:2014 number:1-2 day:30 month:10 pages:231-245 https://dx.doi.org/10.1007/s00153-014-0409-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_206 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 54 2014 1-2 30 10 231-245 |
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10.1007/s00153-014-0409-4 doi (DE-627)SPR001300679 (SPR)s00153-014-0409-4-e DE-627 ger DE-627 rakwb eng Medini, Andrea verfasserin aut Between Polish and completely Baire 2014 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer-Verlag Berlin Heidelberg 2014 Abstract All spaces are assumed to be separable and metrizable. Consider the following properties of a space X. X is Polish.For every countable crowded Q⊆X%${Q \subseteq X}%$ there exists a crowded Q′⊆Q%${Q'\subseteq Q}%$ with compact closure.Every closed subspace of X is either scattered or it contains a homeomorphic copy of 2ω%${2^\omega}%$.Every closed subspace of X is a Baire space. While (4) is the well-known property of being completely Baire, properties (2) and (3) have been recently introduced by Kunen, Medini and Zdomskyy, who named them the Miller property and the Cantor-Bendixson property respectively. It turns out that the implications %${(1) \rightarrow (2) \rightarrow (3) \rightarrow (4)}%$ hold for every space X. Furthermore, it follows from a classical result of Hurewicz that all these implications are equivalences if X is coanalytic. Under the axiom of Projective Determinacy, this equivalence result extends to all projective spaces. We will complete the picture by giving a %${\sf ZFC}%$ counterexample and a consistent definable counterexample of lowest possible complexity to the implication %${(i) \leftarrow (i + 1)}%$ for %${i = 1, 2, 3}%$. For one of these counterexamples we will need a classical theorem of Martin and Solovay, of which we give a new proof, based on a result of Baldwin and Beaudoin. Finally, using a method of Fischer and Friedman, we will investigate how changing the value of the continuum affects the definability of these counterexamples. Along the way, we will show that every uncountable completely Baire space has size continuum. Polish (dpeaa)DE-He213 Miller property (dpeaa)DE-He213 Cantor-Bendixson property (dpeaa)DE-He213 Completely Baire (dpeaa)DE-He213 Hereditarily Baire (dpeaa)DE-He213 Miller-measurable (dpeaa)DE-He213 Zdomskyy, Lyubomyr aut Enthalten in Archive for mathematical logic Berlin : Springer, 1950 54(2014), 1-2 vom: 30. Okt., Seite 231-245 (DE-627)235503215 (DE-600)1398309-X 1432-0665 nnns volume:54 year:2014 number:1-2 day:30 month:10 pages:231-245 https://dx.doi.org/10.1007/s00153-014-0409-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_206 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 54 2014 1-2 30 10 231-245 |
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Consider the following properties of a space X. X is Polish.For every countable crowded Q⊆X%${Q \subseteq X}%$ there exists a crowded Q′⊆Q%${Q'\subseteq Q}%$ with compact closure.Every closed subspace of X is either scattered or it contains a homeomorphic copy of 2ω%${2^\omega}%$.Every closed subspace of X is a Baire space. While (4) is the well-known property of being completely Baire, properties (2) and (3) have been recently introduced by Kunen, Medini and Zdomskyy, who named them the Miller property and the Cantor-Bendixson property respectively. It turns out that the implications %${(1) \rightarrow (2) \rightarrow (3) \rightarrow (4)}%$ hold for every space X. Furthermore, it follows from a classical result of Hurewicz that all these implications are equivalences if X is coanalytic. Under the axiom of Projective Determinacy, this equivalence result extends to all projective spaces. We will complete the picture by giving a %${\sf ZFC}%$ counterexample and a consistent definable counterexample of lowest possible complexity to the implication %${(i) \leftarrow (i + 1)}%$ for %${i = 1, 2, 3}%$. For one of these counterexamples we will need a classical theorem of Martin and Solovay, of which we give a new proof, based on a result of Baldwin and Beaudoin. Finally, using a method of Fischer and Friedman, we will investigate how changing the value of the continuum affects the definability of these counterexamples. 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Between Polish and completely Baire |
| abstract |
Abstract All spaces are assumed to be separable and metrizable. Consider the following properties of a space X. X is Polish.For every countable crowded Q⊆X%${Q \subseteq X}%$ there exists a crowded Q′⊆Q%${Q'\subseteq Q}%$ with compact closure.Every closed subspace of X is either scattered or it contains a homeomorphic copy of 2ω%${2^\omega}%$.Every closed subspace of X is a Baire space. While (4) is the well-known property of being completely Baire, properties (2) and (3) have been recently introduced by Kunen, Medini and Zdomskyy, who named them the Miller property and the Cantor-Bendixson property respectively. It turns out that the implications %${(1) \rightarrow (2) \rightarrow (3) \rightarrow (4)}%$ hold for every space X. Furthermore, it follows from a classical result of Hurewicz that all these implications are equivalences if X is coanalytic. Under the axiom of Projective Determinacy, this equivalence result extends to all projective spaces. We will complete the picture by giving a %${\sf ZFC}%$ counterexample and a consistent definable counterexample of lowest possible complexity to the implication %${(i) \leftarrow (i + 1)}%$ for %${i = 1, 2, 3}%$. For one of these counterexamples we will need a classical theorem of Martin and Solovay, of which we give a new proof, based on a result of Baldwin and Beaudoin. Finally, using a method of Fischer and Friedman, we will investigate how changing the value of the continuum affects the definability of these counterexamples. Along the way, we will show that every uncountable completely Baire space has size continuum. © Springer-Verlag Berlin Heidelberg 2014 |
| abstractGer |
Abstract All spaces are assumed to be separable and metrizable. Consider the following properties of a space X. X is Polish.For every countable crowded Q⊆X%${Q \subseteq X}%$ there exists a crowded Q′⊆Q%${Q'\subseteq Q}%$ with compact closure.Every closed subspace of X is either scattered or it contains a homeomorphic copy of 2ω%${2^\omega}%$.Every closed subspace of X is a Baire space. While (4) is the well-known property of being completely Baire, properties (2) and (3) have been recently introduced by Kunen, Medini and Zdomskyy, who named them the Miller property and the Cantor-Bendixson property respectively. It turns out that the implications %${(1) \rightarrow (2) \rightarrow (3) \rightarrow (4)}%$ hold for every space X. Furthermore, it follows from a classical result of Hurewicz that all these implications are equivalences if X is coanalytic. Under the axiom of Projective Determinacy, this equivalence result extends to all projective spaces. We will complete the picture by giving a %${\sf ZFC}%$ counterexample and a consistent definable counterexample of lowest possible complexity to the implication %${(i) \leftarrow (i + 1)}%$ for %${i = 1, 2, 3}%$. For one of these counterexamples we will need a classical theorem of Martin and Solovay, of which we give a new proof, based on a result of Baldwin and Beaudoin. Finally, using a method of Fischer and Friedman, we will investigate how changing the value of the continuum affects the definability of these counterexamples. Along the way, we will show that every uncountable completely Baire space has size continuum. © Springer-Verlag Berlin Heidelberg 2014 |
| abstract_unstemmed |
Abstract All spaces are assumed to be separable and metrizable. Consider the following properties of a space X. X is Polish.For every countable crowded Q⊆X%${Q \subseteq X}%$ there exists a crowded Q′⊆Q%${Q'\subseteq Q}%$ with compact closure.Every closed subspace of X is either scattered or it contains a homeomorphic copy of 2ω%${2^\omega}%$.Every closed subspace of X is a Baire space. While (4) is the well-known property of being completely Baire, properties (2) and (3) have been recently introduced by Kunen, Medini and Zdomskyy, who named them the Miller property and the Cantor-Bendixson property respectively. It turns out that the implications %${(1) \rightarrow (2) \rightarrow (3) \rightarrow (4)}%$ hold for every space X. Furthermore, it follows from a classical result of Hurewicz that all these implications are equivalences if X is coanalytic. Under the axiom of Projective Determinacy, this equivalence result extends to all projective spaces. We will complete the picture by giving a %${\sf ZFC}%$ counterexample and a consistent definable counterexample of lowest possible complexity to the implication %${(i) \leftarrow (i + 1)}%$ for %${i = 1, 2, 3}%$. For one of these counterexamples we will need a classical theorem of Martin and Solovay, of which we give a new proof, based on a result of Baldwin and Beaudoin. Finally, using a method of Fischer and Friedman, we will investigate how changing the value of the continuum affects the definability of these counterexamples. Along the way, we will show that every uncountable completely Baire space has size continuum. © Springer-Verlag Berlin Heidelberg 2014 |
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| container_issue |
1-2 |
| title_short |
Between Polish and completely Baire |
| url |
https://dx.doi.org/10.1007/s00153-014-0409-4 |
| remote_bool |
true |
| author2 |
Zdomskyy, Lyubomyr |
| author2Str |
Zdomskyy, Lyubomyr |
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c |
| isOA_txt |
false |
| hochschulschrift_bool |
false |
| doi_str |
10.1007/s00153-014-0409-4 |
| up_date |
2024-07-03T21:38:29.974Z |
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1803595504659988480 |
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| score |
7.4000187 |