Mixing
Union and intersection of $F_sigma$ sets [closed]
$begingroup$
A set $A ⊆ R$ is called an $F_sigma$ set if it can be written as the
countable union of closed sets. A set $B ⊆ R$ is called a $G_δ$ set if it can be
written as the countable intersection of open sets.
How to prove, that:
- The countable union of $F_σ$ sets is an $F_σ$ set.
- The finite intersection of $F_σ$ sets is an $F_σ$ set.
Thanks for help!
general-topology analysis
edited Feb 2 at 23:33
Henno Brandsma
116k349127
asked Feb 2 at 18:15
RonaldRonald
101
$endgroup$
closed as off-topic by Paul Frost, YiFan, Cesareo, José Carlos Santos, mrtaurho Feb 3 at 13:41
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Paul Frost, YiFan, Cesareo, José Carlos Santos, mrtaurho
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
A set $A ⊆ R$ is called an $F_sigma$ set if it can be written as the
countable union of closed sets. A set $B ⊆ R$ is called a $G_δ$ set if it can be
written as the countable intersection of open sets.
How to prove, that:
- The countable union of $F_σ$ sets is an $F_σ$ set.
- The finite intersection of $F_σ$ sets is an $F_σ$ set.
Thanks for help!
general-topology analysis
edited Feb 2 at 23:33
Henno Brandsma
116k349127
asked Feb 2 at 18:15
RonaldRonald
101
$endgroup$
closed as off-topic by Paul Frost, YiFan, Cesareo, José Carlos Santos, mrtaurho Feb 3 at 13:41
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Paul Frost, YiFan, Cesareo, José Carlos Santos, mrtaurho
If this question can be reworded to fit the rules in the help center, please edit the question.
1
$begingroup$
What is $bigcup_{i=1}^{n}F_sigma $ written out?
$endgroup$
– babemcnuggets
Feb 2 at 18:23
$begingroup$
The first claim is obvious since countable union of countable unions of closed sets is again a countable union of closed sets (union is associative). For the second claim, this and finite induction does the job.
$endgroup$
– learner
Feb 2 at 18:24
add a comment |
$begingroup$
A set $A ⊆ R$ is called an $F_sigma$ set if it can be written as the
countable union of closed sets. A set $B ⊆ R$ is called a $G_δ$ set if it can be
written as the countable intersection of open sets.
How to prove, that:
- The countable union of $F_σ$ sets is an $F_σ$ set.
- The finite intersection of $F_σ$ sets is an $F_σ$ set.
Thanks for help!
general-topology analysis
edited Feb 2 at 23:33
Henno Brandsma
116k349127
asked Feb 2 at 18:15
RonaldRonald
101
$endgroup$
A set $A ⊆ R$ is called an $F_sigma$ set if it can be written as the
countable union of closed sets. A set $B ⊆ R$ is called a $G_δ$ set if it can be
written as the countable intersection of open sets.
How to prove, that:
- The countable union of $F_σ$ sets is an $F_σ$ set.
- The finite intersection of $F_σ$ sets is an $F_σ$ set.
Thanks for help!
general-topology analysis
general-topology analysis
edited Feb 2 at 23:33
Henno Brandsma
116k349127
asked Feb 2 at 18:15
RonaldRonald
101
edited Feb 2 at 23:33
Henno Brandsma
116k349127
asked Feb 2 at 18:15
RonaldRonald
101
edited Feb 2 at 23:33
Henno Brandsma
116k349127
edited Feb 2 at 23:33
Henno Brandsma
116k349127
edited Feb 2 at 23:33
Henno Brandsma
116k349127
116k349127
asked Feb 2 at 18:15
RonaldRonald
101
asked Feb 2 at 18:15
RonaldRonald
101
asked Feb 2 at 18:15
RonaldRonald
101
101
closed as off-topic by Paul Frost, YiFan, Cesareo, José Carlos Santos, mrtaurho Feb 3 at 13:41
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Paul Frost, YiFan, Cesareo, José Carlos Santos, mrtaurho
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Paul Frost, YiFan, Cesareo, José Carlos Santos, mrtaurho Feb 3 at 13:41
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Paul Frost, YiFan, Cesareo, José Carlos Santos, mrtaurho
If this question can be reworded to fit the rules in the help center, please edit the question.
1
$begingroup$
What is $bigcup_{i=1}^{n}F_sigma $ written out?
$endgroup$
– babemcnuggets
Feb 2 at 18:23
$begingroup$
The first claim is obvious since countable union of countable unions of closed sets is again a countable union of closed sets (union is associative). For the second claim, this and finite induction does the job.
$endgroup$
– learner
Feb 2 at 18:24
add a comment |
1
$begingroup$
What is $bigcup_{i=1}^{n}F_sigma $ written out?
$endgroup$
– babemcnuggets
Feb 2 at 18:23
$begingroup$
The first claim is obvious since countable union of countable unions of closed sets is again a countable union of closed sets (union is associative). For the second claim, this and finite induction does the job.
$endgroup$
– learner
Feb 2 at 18:24
1
1
$begingroup$
What is $bigcup_{i=1}^{n}F_sigma $ written out?
$endgroup$
– babemcnuggets
Feb 2 at 18:23
$begingroup$
What is $bigcup_{i=1}^{n}F_sigma $ written out?
$endgroup$
– babemcnuggets
Feb 2 at 18:23
$begingroup$
The first claim is obvious since countable union of countable unions of closed sets is again a countable union of closed sets (union is associative). For the second claim, this and finite induction does the job.
$endgroup$
– learner
Feb 2 at 18:24
$begingroup$
The first claim is obvious since countable union of countable unions of closed sets is again a countable union of closed sets (union is associative). For the second claim, this and finite induction does the job.
$endgroup$
– learner
Feb 2 at 18:24
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Suppose $P_n, n in mathbb{N}$ are $F_sigma$ sets so we can write $$P_n = bigcup_{m in mathbb{N}} F_{n,m},text{ with } F_{n,m} text{ closed in} X$$
But then $$bigcup_{n in mathbb{N}} P_n = bigcup_{(n,m) in mathbb{N} times mathbb{N}} F_{n,m}$$
and the union is also a countable (as $mathbb{N} times mathbb{N}$ is countable) union of closed sets, hence an $F_sigma$.
Also $$P_1 cap P_2 = bigcup_{(n,m) in mathbb{N} times mathbb{N}} (F_{1,n} cap F_{2,m})$$
(check this identity), so the intersection of two $F_sigma$'s is an $F_sigma$ again, and by induction we can extend this to all finite intersections in the standard way, or alternatively use that
$$bigcap_{i=1}^k P_i = bigcup_{f in mathbb{N}^k} bigcap_{n=1}^k F_{n,f(n)}$$ as a generalisation of the previous identity.
answered Feb 2 at 23:33
Henno BrandsmaHenno Brandsma
116k349127
$endgroup$
$begingroup$
Can i argue in the same way, that countable intersection of $F_sigma$ sets is an $F_sigma$ set?
$endgroup$
– Ronald
Feb 3 at 12:35
$begingroup$
@Ronald, no because the countable intersection involves the union of continuum many intersections if we write it in the same way: $bigcap_n P_n = bigcup_{f in mathbb{N}^mathbb{N}} bigcap_{n in mathbb{N}} F_{n, f(n)}$. We need infinite sequences of integers instead of finite ones and that makes a big difference.
$endgroup$
– Henno Brandsma
Feb 3 at 12:40
$begingroup$
@Ronald and one can show there are $(F_sigma)_delta$ sets that are not $F_sigma$, but this gets tricky. Believe it: it's is false. A whole theory called "descriptive set theory" has been developed for such questions. See e.g. the book by Kechris of that title if you really want to know more about this.
$endgroup$
– Henno Brandsma
Feb 3 at 12:44
$begingroup$
Thanks, you really help me!!
$endgroup$
– Ronald
Feb 3 at 13:02
$begingroup$
@Ronald glad you've been helped. You're welcome.
$endgroup$
– Henno Brandsma
Feb 3 at 13:03
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Suppose $P_n, n in mathbb{N}$ are $F_sigma$ sets so we can write $$P_n = bigcup_{m in mathbb{N}} F_{n,m},text{ with } F_{n,m} text{ closed in} X$$
But then $$bigcup_{n in mathbb{N}} P_n = bigcup_{(n,m) in mathbb{N} times mathbb{N}} F_{n,m}$$
and the union is also a countable (as $mathbb{N} times mathbb{N}$ is countable) union of closed sets, hence an $F_sigma$.
Also $$P_1 cap P_2 = bigcup_{(n,m) in mathbb{N} times mathbb{N}} (F_{1,n} cap F_{2,m})$$
(check this identity), so the intersection of two $F_sigma$'s is an $F_sigma$ again, and by induction we can extend this to all finite intersections in the standard way, or alternatively use that
$$bigcap_{i=1}^k P_i = bigcup_{f in mathbb{N}^k} bigcap_{n=1}^k F_{n,f(n)}$$ as a generalisation of the previous identity.
answered Feb 2 at 23:33
Henno BrandsmaHenno Brandsma
116k349127
$endgroup$
$begingroup$
Can i argue in the same way, that countable intersection of $F_sigma$ sets is an $F_sigma$ set?
$endgroup$
– Ronald
Feb 3 at 12:35
$begingroup$
@Ronald, no because the countable intersection involves the union of continuum many intersections if we write it in the same way: $bigcap_n P_n = bigcup_{f in mathbb{N}^mathbb{N}} bigcap_{n in mathbb{N}} F_{n, f(n)}$. We need infinite sequences of integers instead of finite ones and that makes a big difference.
$endgroup$
– Henno Brandsma
Feb 3 at 12:40
$begingroup$
@Ronald and one can show there are $(F_sigma)_delta$ sets that are not $F_sigma$, but this gets tricky. Believe it: it's is false. A whole theory called "descriptive set theory" has been developed for such questions. See e.g. the book by Kechris of that title if you really want to know more about this.
$endgroup$
– Henno Brandsma
Feb 3 at 12:44
$begingroup$
Thanks, you really help me!!
$endgroup$
– Ronald
Feb 3 at 13:02
$begingroup$
@Ronald glad you've been helped. You're welcome.
$endgroup$
– Henno Brandsma
Feb 3 at 13:03
add a comment |
$begingroup$
Suppose $P_n, n in mathbb{N}$ are $F_sigma$ sets so we can write $$P_n = bigcup_{m in mathbb{N}} F_{n,m},text{ with } F_{n,m} text{ closed in} X$$
But then $$bigcup_{n in mathbb{N}} P_n = bigcup_{(n,m) in mathbb{N} times mathbb{N}} F_{n,m}$$
and the union is also a countable (as $mathbb{N} times mathbb{N}$ is countable) union of closed sets, hence an $F_sigma$.
Also $$P_1 cap P_2 = bigcup_{(n,m) in mathbb{N} times mathbb{N}} (F_{1,n} cap F_{2,m})$$
(check this identity), so the intersection of two $F_sigma$'s is an $F_sigma$ again, and by induction we can extend this to all finite intersections in the standard way, or alternatively use that
$$bigcap_{i=1}^k P_i = bigcup_{f in mathbb{N}^k} bigcap_{n=1}^k F_{n,f(n)}$$ as a generalisation of the previous identity.
answered Feb 2 at 23:33
Henno BrandsmaHenno Brandsma
116k349127
$endgroup$
$begingroup$
Can i argue in the same way, that countable intersection of $F_sigma$ sets is an $F_sigma$ set?
$endgroup$
– Ronald
Feb 3 at 12:35
$begingroup$
@Ronald, no because the countable intersection involves the union of continuum many intersections if we write it in the same way: $bigcap_n P_n = bigcup_{f in mathbb{N}^mathbb{N}} bigcap_{n in mathbb{N}} F_{n, f(n)}$. We need infinite sequences of integers instead of finite ones and that makes a big difference.
$endgroup$
– Henno Brandsma
Feb 3 at 12:40
$begingroup$
@Ronald and one can show there are $(F_sigma)_delta$ sets that are not $F_sigma$, but this gets tricky. Believe it: it's is false. A whole theory called "descriptive set theory" has been developed for such questions. See e.g. the book by Kechris of that title if you really want to know more about this.
$endgroup$
– Henno Brandsma
Feb 3 at 12:44
$begingroup$
Thanks, you really help me!!
$endgroup$
– Ronald
Feb 3 at 13:02
$begingroup$
@Ronald glad you've been helped. You're welcome.
$endgroup$
– Henno Brandsma
Feb 3 at 13:03
add a comment |
$begingroup$
Suppose $P_n, n in mathbb{N}$ are $F_sigma$ sets so we can write $$P_n = bigcup_{m in mathbb{N}} F_{n,m},text{ with } F_{n,m} text{ closed in} X$$
But then $$bigcup_{n in mathbb{N}} P_n = bigcup_{(n,m) in mathbb{N} times mathbb{N}} F_{n,m}$$
and the union is also a countable (as $mathbb{N} times mathbb{N}$ is countable) union of closed sets, hence an $F_sigma$.
Also $$P_1 cap P_2 = bigcup_{(n,m) in mathbb{N} times mathbb{N}} (F_{1,n} cap F_{2,m})$$
(check this identity), so the intersection of two $F_sigma$'s is an $F_sigma$ again, and by induction we can extend this to all finite intersections in the standard way, or alternatively use that
$$bigcap_{i=1}^k P_i = bigcup_{f in mathbb{N}^k} bigcap_{n=1}^k F_{n,f(n)}$$ as a generalisation of the previous identity.
answered Feb 2 at 23:33
Henno BrandsmaHenno Brandsma
116k349127
$endgroup$
Suppose $P_n, n in mathbb{N}$ are $F_sigma$ sets so we can write $$P_n = bigcup_{m in mathbb{N}} F_{n,m},text{ with } F_{n,m} text{ closed in} X$$
But then $$bigcup_{n in mathbb{N}} P_n = bigcup_{(n,m) in mathbb{N} times mathbb{N}} F_{n,m}$$
and the union is also a countable (as $mathbb{N} times mathbb{N}$ is countable) union of closed sets, hence an $F_sigma$.
Also $$P_1 cap P_2 = bigcup_{(n,m) in mathbb{N} times mathbb{N}} (F_{1,n} cap F_{2,m})$$
(check this identity), so the intersection of two $F_sigma$'s is an $F_sigma$ again, and by induction we can extend this to all finite intersections in the standard way, or alternatively use that
$$bigcap_{i=1}^k P_i = bigcup_{f in mathbb{N}^k} bigcap_{n=1}^k F_{n,f(n)}$$ as a generalisation of the previous identity.
answered Feb 2 at 23:33
Henno BrandsmaHenno Brandsma
116k349127
edited Feb 3 at 12:47
answered Feb 2 at 23:33
Henno BrandsmaHenno Brandsma
116k349127
answered Feb 2 at 23:33
Henno BrandsmaHenno Brandsma
116k349127
answered Feb 2 at 23:33
Henno BrandsmaHenno Brandsma
116k349127
116k349127
$begingroup$
Can i argue in the same way, that countable intersection of $F_sigma$ sets is an $F_sigma$ set?
$endgroup$
– Ronald
Feb 3 at 12:35
$begingroup$
@Ronald, no because the countable intersection involves the union of continuum many intersections if we write it in the same way: $bigcap_n P_n = bigcup_{f in mathbb{N}^mathbb{N}} bigcap_{n in mathbb{N}} F_{n, f(n)}$. We need infinite sequences of integers instead of finite ones and that makes a big difference.
$endgroup$
– Henno Brandsma
Feb 3 at 12:40
$begingroup$
@Ronald and one can show there are $(F_sigma)_delta$ sets that are not $F_sigma$, but this gets tricky. Believe it: it's is false. A whole theory called "descriptive set theory" has been developed for such questions. See e.g. the book by Kechris of that title if you really want to know more about this.
$endgroup$
– Henno Brandsma
Feb 3 at 12:44
$begingroup$
Thanks, you really help me!!
$endgroup$
– Ronald
Feb 3 at 13:02
$begingroup$
@Ronald glad you've been helped. You're welcome.
$endgroup$
– Henno Brandsma
Feb 3 at 13:03
add a comment |
$begingroup$
Can i argue in the same way, that countable intersection of $F_sigma$ sets is an $F_sigma$ set?
$endgroup$
– Ronald
Feb 3 at 12:35
$begingroup$
@Ronald, no because the countable intersection involves the union of continuum many intersections if we write it in the same way: $bigcap_n P_n = bigcup_{f in mathbb{N}^mathbb{N}} bigcap_{n in mathbb{N}} F_{n, f(n)}$. We need infinite sequences of integers instead of finite ones and that makes a big difference.
$endgroup$
– Henno Brandsma
Feb 3 at 12:40
$begingroup$
@Ronald and one can show there are $(F_sigma)_delta$ sets that are not $F_sigma$, but this gets tricky. Believe it: it's is false. A whole theory called "descriptive set theory" has been developed for such questions. See e.g. the book by Kechris of that title if you really want to know more about this.
$endgroup$
– Henno Brandsma
Feb 3 at 12:44
$begingroup$
Thanks, you really help me!!
$endgroup$
– Ronald
Feb 3 at 13:02
$begingroup$
@Ronald glad you've been helped. You're welcome.
$endgroup$
– Henno Brandsma
Feb 3 at 13:03
$begingroup$
Can i argue in the same way, that countable intersection of $F_sigma$ sets is an $F_sigma$ set?
$endgroup$
– Ronald
Feb 3 at 12:35
$begingroup$
Can i argue in the same way, that countable intersection of $F_sigma$ sets is an $F_sigma$ set?
$endgroup$
– Ronald
Feb 3 at 12:35
$begingroup$
@Ronald, no because the countable intersection involves the union of continuum many intersections if we write it in the same way: $bigcap_n P_n = bigcup_{f in mathbb{N}^mathbb{N}} bigcap_{n in mathbb{N}} F_{n, f(n)}$. We need infinite sequences of integers instead of finite ones and that makes a big difference.
$endgroup$
– Henno Brandsma
Feb 3 at 12:40
$begingroup$
@Ronald, no because the countable intersection involves the union of continuum many intersections if we write it in the same way: $bigcap_n P_n = bigcup_{f in mathbb{N}^mathbb{N}} bigcap_{n in mathbb{N}} F_{n, f(n)}$. We need infinite sequences of integers instead of finite ones and that makes a big difference.
$endgroup$
– Henno Brandsma
Feb 3 at 12:40
$begingroup$
@Ronald and one can show there are $(F_sigma)_delta$ sets that are not $F_sigma$, but this gets tricky. Believe it: it's is false. A whole theory called "descriptive set theory" has been developed for such questions. See e.g. the book by Kechris of that title if you really want to know more about this.
$endgroup$
– Henno Brandsma
Feb 3 at 12:44
$begingroup$
@Ronald and one can show there are $(F_sigma)_delta$ sets that are not $F_sigma$, but this gets tricky. Believe it: it's is false. A whole theory called "descriptive set theory" has been developed for such questions. See e.g. the book by Kechris of that title if you really want to know more about this.
$endgroup$
– Henno Brandsma
Feb 3 at 12:44
$begingroup$
Thanks, you really help me!!
$endgroup$
– Ronald
Feb 3 at 13:02
$begingroup$
Thanks, you really help me!!
$endgroup$
– Ronald
Feb 3 at 13:02
$begingroup$
@Ronald glad you've been helped. You're welcome.
$endgroup$
– Henno Brandsma
Feb 3 at 13:03
$begingroup$
@Ronald glad you've been helped. You're welcome.
$endgroup$
– Henno Brandsma
Feb 3 at 13:03
add a comment |
1
$begingroup$
What is $bigcup_{i=1}^{n}F_sigma $ written out?
$endgroup$
– babemcnuggets
Feb 2 at 18:23
$begingroup$
The first claim is obvious since countable union of countable unions of closed sets is again a countable union of closed sets (union is associative). For the second claim, this and finite induction does the job.
$endgroup$
– learner
Feb 2 at 18:24