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Baire Property on Baire Spaces (Kechris' book)
Concrete example of the property of Baire on a finite discrete spaceQuestion about of Baire property and Baire spaceThe inverse image of dense set is dense and of a comeager set is comeager?.Every comeager set in a perfect Polish space contains an uncountable dense $G_delta$ setQuestion about comeager set in a Polish spaceAnalytic sets have perfect set property (Kechris)Continuous image of a Polish space to another has the Baire propertyIs a weak basis of a topological space a basis and vice versa? (Kechris' book)Theorem $(8.29)$ (Kechris)The canonical open set which is equal to a set with the BP modulo meager sets is regular open (Kechris)
$begingroup$
Kechris states the following result in "Classical Descriptive Set Theory", pp. $48$:
($boldsymbol8.26$) Proposition
Let $X$ be a topological space and suppose that $Asubseteq X$ has the BP (Baire Property).
Then either $A$ is meager or there is a nonempty open set $Usubseteq X$ s.t. $A$ is comeager in $U$.
If $X$ is a Baire space, exactly one of these alternatives holds.
Well, I'm not able to prove the last assertion. I've worked on it for some time and maybe I can't something different from what I have already tried (I think the proof is very elementary).
For the convenience of the reader, I recall that
$A$ has the BP iff there is an open set $Usubseteq X$ s.t. the symmetric difference $Atriangle U$ is meager;
$X$ is a Baire space If it satisfies one of the following equivalent conditions:
(i) every nonempty open set in $X$ is non-meager;
(ii) every comeager set in $X$ is dense;
(iii) the intersection of countably many dense open sets on $X$ is dense.
Thank you in advance for your help.
general-topology descriptive-set-theory
edited Apr 2 at 8:28
Henno Brandsma
115k349125
asked Mar 29 at 12:25
LBJFSLBJFS
367112
$endgroup$
add a comment |
$begingroup$
Kechris states the following result in "Classical Descriptive Set Theory", pp. $48$:
($boldsymbol8.26$) Proposition
Let $X$ be a topological space and suppose that $Asubseteq X$ has the BP (Baire Property).
Then either $A$ is meager or there is a nonempty open set $Usubseteq X$ s.t. $A$ is comeager in $U$.
If $X$ is a Baire space, exactly one of these alternatives holds.
Well, I'm not able to prove the last assertion. I've worked on it for some time and maybe I can't something different from what I have already tried (I think the proof is very elementary).
For the convenience of the reader, I recall that
$A$ has the BP iff there is an open set $Usubseteq X$ s.t. the symmetric difference $Atriangle U$ is meager;
$X$ is a Baire space If it satisfies one of the following equivalent conditions:
(i) every nonempty open set in $X$ is non-meager;
(ii) every comeager set in $X$ is dense;
(iii) the intersection of countably many dense open sets on $X$ is dense.
Thank you in advance for your help.
general-topology descriptive-set-theory
edited Apr 2 at 8:28
Henno Brandsma
115k349125
asked Mar 29 at 12:25
LBJFSLBJFS
367112
$endgroup$
add a comment |
$begingroup$
Kechris states the following result in "Classical Descriptive Set Theory", pp. $48$:
($boldsymbol8.26$) Proposition
Let $X$ be a topological space and suppose that $Asubseteq X$ has the BP (Baire Property).
Then either $A$ is meager or there is a nonempty open set $Usubseteq X$ s.t. $A$ is comeager in $U$.
If $X$ is a Baire space, exactly one of these alternatives holds.
Well, I'm not able to prove the last assertion. I've worked on it for some time and maybe I can't something different from what I have already tried (I think the proof is very elementary).
For the convenience of the reader, I recall that
$A$ has the BP iff there is an open set $Usubseteq X$ s.t. the symmetric difference $Atriangle U$ is meager;
$X$ is a Baire space If it satisfies one of the following equivalent conditions:
(i) every nonempty open set in $X$ is non-meager;
(ii) every comeager set in $X$ is dense;
(iii) the intersection of countably many dense open sets on $X$ is dense.
Thank you in advance for your help.
general-topology descriptive-set-theory
edited Apr 2 at 8:28
Henno Brandsma
115k349125
asked Mar 29 at 12:25
LBJFSLBJFS
367112
$endgroup$
Kechris states the following result in "Classical Descriptive Set Theory", pp. $48$:
($boldsymbol8.26$) Proposition
Let $X$ be a topological space and suppose that $Asubseteq X$ has the BP (Baire Property).
Then either $A$ is meager or there is a nonempty open set $Usubseteq X$ s.t. $A$ is comeager in $U$.
If $X$ is a Baire space, exactly one of these alternatives holds.
Well, I'm not able to prove the last assertion. I've worked on it for some time and maybe I can't something different from what I have already tried (I think the proof is very elementary).
For the convenience of the reader, I recall that
$A$ has the BP iff there is an open set $Usubseteq X$ s.t. the symmetric difference $Atriangle U$ is meager;
$X$ is a Baire space If it satisfies one of the following equivalent conditions:
(i) every nonempty open set in $X$ is non-meager;
(ii) every comeager set in $X$ is dense;
(iii) the intersection of countably many dense open sets on $X$ is dense.
Thank you in advance for your help.
general-topology descriptive-set-theory
general-topology descriptive-set-theory
edited Apr 2 at 8:28
Henno Brandsma
115k349125
asked Mar 29 at 12:25
LBJFSLBJFS
367112
edited Apr 2 at 8:28
Henno Brandsma
115k349125
asked Mar 29 at 12:25
LBJFSLBJFS
367112
edited Apr 2 at 8:28
Henno Brandsma
115k349125
edited Apr 2 at 8:28
Henno Brandsma
115k349125
edited Apr 2 at 8:28
Henno Brandsma
115k349125
115k349125
asked Mar 29 at 12:25
LBJFSLBJFS
367112
asked Mar 29 at 12:25
LBJFSLBJFS
367112
asked Mar 29 at 12:25
LBJFSLBJFS
367112
367112
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
If you've already proved the first assertion, then the second assertion just says that a meager set $A$ can't be comeager in a nonempty open set $U$. Well, suppose $A$ is meager, $U$ is nonempty and open, and $U-A$ is meager. Then $U$, being included in the union of two meager sets $A$ and $U-A$, is meager, contrary to the assumption that $X$ is a Baire space.
answered Mar 29 at 16:00
Andreas BlassAndreas Blass
50.5k452109
$endgroup$
$begingroup$
I thought that it was simple, but this is a great blunder! I would like to thank you for this proof... I was in trouble at that moment :)
$endgroup$
– LBJFS
Mar 29 at 19:02
add a comment |
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1 Answer
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$begingroup$
If you've already proved the first assertion, then the second assertion just says that a meager set $A$ can't be comeager in a nonempty open set $U$. Well, suppose $A$ is meager, $U$ is nonempty and open, and $U-A$ is meager. Then $U$, being included in the union of two meager sets $A$ and $U-A$, is meager, contrary to the assumption that $X$ is a Baire space.
answered Mar 29 at 16:00
Andreas BlassAndreas Blass
50.5k452109
$endgroup$
$begingroup$
I thought that it was simple, but this is a great blunder! I would like to thank you for this proof... I was in trouble at that moment :)
$endgroup$
– LBJFS
Mar 29 at 19:02
add a comment |
$begingroup$
If you've already proved the first assertion, then the second assertion just says that a meager set $A$ can't be comeager in a nonempty open set $U$. Well, suppose $A$ is meager, $U$ is nonempty and open, and $U-A$ is meager. Then $U$, being included in the union of two meager sets $A$ and $U-A$, is meager, contrary to the assumption that $X$ is a Baire space.
answered Mar 29 at 16:00
Andreas BlassAndreas Blass
50.5k452109
$endgroup$
$begingroup$
I thought that it was simple, but this is a great blunder! I would like to thank you for this proof... I was in trouble at that moment :)
$endgroup$
– LBJFS
Mar 29 at 19:02
add a comment |
$begingroup$
If you've already proved the first assertion, then the second assertion just says that a meager set $A$ can't be comeager in a nonempty open set $U$. Well, suppose $A$ is meager, $U$ is nonempty and open, and $U-A$ is meager. Then $U$, being included in the union of two meager sets $A$ and $U-A$, is meager, contrary to the assumption that $X$ is a Baire space.
answered Mar 29 at 16:00
Andreas BlassAndreas Blass
50.5k452109
$endgroup$
If you've already proved the first assertion, then the second assertion just says that a meager set $A$ can't be comeager in a nonempty open set $U$. Well, suppose $A$ is meager, $U$ is nonempty and open, and $U-A$ is meager. Then $U$, being included in the union of two meager sets $A$ and $U-A$, is meager, contrary to the assumption that $X$ is a Baire space.
answered Mar 29 at 16:00
Andreas BlassAndreas Blass
50.5k452109
answered Mar 29 at 16:00
Andreas BlassAndreas Blass
50.5k452109
answered Mar 29 at 16:00
Andreas BlassAndreas Blass
50.5k452109
answered Mar 29 at 16:00
Andreas BlassAndreas Blass
50.5k452109
50.5k452109
$begingroup$
I thought that it was simple, but this is a great blunder! I would like to thank you for this proof... I was in trouble at that moment :)
$endgroup$
– LBJFS
Mar 29 at 19:02
add a comment |
$begingroup$
I thought that it was simple, but this is a great blunder! I would like to thank you for this proof... I was in trouble at that moment :)
$endgroup$
– LBJFS
Mar 29 at 19:02
$begingroup$
I thought that it was simple, but this is a great blunder! I would like to thank you for this proof... I was in trouble at that moment :)
$endgroup$
– LBJFS
Mar 29 at 19:02
$begingroup$
I thought that it was simple, but this is a great blunder! I would like to thank you for this proof... I was in trouble at that moment :)
$endgroup$
– LBJFS
Mar 29 at 19:02
add a comment |
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