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
Autor*in: |
Medini, Andrea [verfasserIn] |
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Artikel |
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Englisch |
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2014 |
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Anmerkung: |
© Springer-Verlag Berlin Heidelberg 2014 |
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Übergeordnetes Werk: |
Enthalten in: Archive for mathematical logic - Springer Berlin Heidelberg, 1988, 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 |
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DOI / URN: |
10.1007/s00153-014-0409-4 |
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Katalog-ID: |
OLC2060249325 |
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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. | ||
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10.1007/s00153-014-0409-4 doi (DE-627)OLC2060249325 (DE-He213)s00153-014-0409-4-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Medini, Andrea verfasserin aut Between Polish and completely Baire 2014 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc 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 Miller property Cantor-Bendixson property Completely Baire Hereditarily Baire Miller-measurable Zdomskyy, Lyubomyr aut Enthalten in Archive for mathematical logic Springer Berlin Heidelberg, 1988 54(2014), 1-2 vom: 30. Okt., Seite 231-245 (DE-627)130412910 (DE-600)623073-8 (DE-576)015915948 0933-5846 nnns volume:54 year:2014 number:1-2 day:30 month:10 pages:231-245 https://doi.org/10.1007/s00153-014-0409-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_11 GBV_ILN_24 GBV_ILN_40 GBV_ILN_70 GBV_ILN_2004 GBV_ILN_2018 GBV_ILN_2088 GBV_ILN_4277 GBV_ILN_4700 AR 54 2014 1-2 30 10 231-245 |
spelling |
10.1007/s00153-014-0409-4 doi (DE-627)OLC2060249325 (DE-He213)s00153-014-0409-4-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Medini, Andrea verfasserin aut Between Polish and completely Baire 2014 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc 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 Miller property Cantor-Bendixson property Completely Baire Hereditarily Baire Miller-measurable Zdomskyy, Lyubomyr aut Enthalten in Archive for mathematical logic Springer Berlin Heidelberg, 1988 54(2014), 1-2 vom: 30. Okt., Seite 231-245 (DE-627)130412910 (DE-600)623073-8 (DE-576)015915948 0933-5846 nnns volume:54 year:2014 number:1-2 day:30 month:10 pages:231-245 https://doi.org/10.1007/s00153-014-0409-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_11 GBV_ILN_24 GBV_ILN_40 GBV_ILN_70 GBV_ILN_2004 GBV_ILN_2018 GBV_ILN_2088 GBV_ILN_4277 GBV_ILN_4700 AR 54 2014 1-2 30 10 231-245 |
allfields_unstemmed |
10.1007/s00153-014-0409-4 doi (DE-627)OLC2060249325 (DE-He213)s00153-014-0409-4-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Medini, Andrea verfasserin aut Between Polish and completely Baire 2014 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc 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 Miller property Cantor-Bendixson property Completely Baire Hereditarily Baire Miller-measurable Zdomskyy, Lyubomyr aut Enthalten in Archive for mathematical logic Springer Berlin Heidelberg, 1988 54(2014), 1-2 vom: 30. Okt., Seite 231-245 (DE-627)130412910 (DE-600)623073-8 (DE-576)015915948 0933-5846 nnns volume:54 year:2014 number:1-2 day:30 month:10 pages:231-245 https://doi.org/10.1007/s00153-014-0409-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_11 GBV_ILN_24 GBV_ILN_40 GBV_ILN_70 GBV_ILN_2004 GBV_ILN_2018 GBV_ILN_2088 GBV_ILN_4277 GBV_ILN_4700 AR 54 2014 1-2 30 10 231-245 |
allfieldsGer |
10.1007/s00153-014-0409-4 doi (DE-627)OLC2060249325 (DE-He213)s00153-014-0409-4-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Medini, Andrea verfasserin aut Between Polish and completely Baire 2014 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc 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 Miller property Cantor-Bendixson property Completely Baire Hereditarily Baire Miller-measurable Zdomskyy, Lyubomyr aut Enthalten in Archive for mathematical logic Springer Berlin Heidelberg, 1988 54(2014), 1-2 vom: 30. Okt., Seite 231-245 (DE-627)130412910 (DE-600)623073-8 (DE-576)015915948 0933-5846 nnns volume:54 year:2014 number:1-2 day:30 month:10 pages:231-245 https://doi.org/10.1007/s00153-014-0409-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_11 GBV_ILN_24 GBV_ILN_40 GBV_ILN_70 GBV_ILN_2004 GBV_ILN_2018 GBV_ILN_2088 GBV_ILN_4277 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)OLC2060249325 (DE-He213)s00153-014-0409-4-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Medini, Andrea verfasserin aut Between Polish and completely Baire 2014 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc 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 Miller property Cantor-Bendixson property Completely Baire Hereditarily Baire Miller-measurable Zdomskyy, Lyubomyr aut Enthalten in Archive for mathematical logic Springer Berlin Heidelberg, 1988 54(2014), 1-2 vom: 30. Okt., Seite 231-245 (DE-627)130412910 (DE-600)623073-8 (DE-576)015915948 0933-5846 nnns volume:54 year:2014 number:1-2 day:30 month:10 pages:231-245 https://doi.org/10.1007/s00153-014-0409-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_11 GBV_ILN_24 GBV_ILN_40 GBV_ILN_70 GBV_ILN_2004 GBV_ILN_2018 GBV_ILN_2088 GBV_ILN_4277 GBV_ILN_4700 AR 54 2014 1-2 30 10 231-245 |
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Between Polish and completely Baire |
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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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