Finding a sequence of sets whose limit is a given set [closed]
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Suppose $B$ is a ball in $mathbb{R}^{n}$ with $n>1$, and $F$ a closed set with empty interior. Is there an increasing sequence of sets $E_{1}subset,...E_{n}subset E_{n+1},...$, all Borel sets, such that each $E_{n}subset Bsetminus F$ but $cup_{n}E_{n}=B$?
real-analysis measure-theory elementary-set-theory
asked Nov 19 at 5:29
M. Rahmat
291211
closed as unclear what you're asking by Kavi Rama Murthy, user90369, user10354138, Shailesh, Cesareo Nov 20 at 1:15
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
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Suppose $B$ is a ball in $mathbb{R}^{n}$ with $n>1$, and $F$ a closed set with empty interior. Is there an increasing sequence of sets $E_{1}subset,...E_{n}subset E_{n+1},...$, all Borel sets, such that each $E_{n}subset Bsetminus F$ but $cup_{n}E_{n}=B$?
real-analysis measure-theory elementary-set-theory
asked Nov 19 at 5:29
M. Rahmat
291211
closed as unclear what you're asking by Kavi Rama Murthy, user90369, user10354138, Shailesh, Cesareo Nov 20 at 1:15
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
4
Not if $Bcap F$ is nonempty. Is this actually the question you wished to ask?
– Lord Shark the Unknown
Nov 19 at 5:32
$F$ is a part of $B$ and so their intersection is not empty.
– M. Rahmat
Nov 19 at 6:16
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Suppose $B$ is a ball in $mathbb{R}^{n}$ with $n>1$, and $F$ a closed set with empty interior. Is there an increasing sequence of sets $E_{1}subset,...E_{n}subset E_{n+1},...$, all Borel sets, such that each $E_{n}subset Bsetminus F$ but $cup_{n}E_{n}=B$?
real-analysis measure-theory elementary-set-theory
asked Nov 19 at 5:29
M. Rahmat
291211
Suppose $B$ is a ball in $mathbb{R}^{n}$ with $n>1$, and $F$ a closed set with empty interior. Is there an increasing sequence of sets $E_{1}subset,...E_{n}subset E_{n+1},...$, all Borel sets, such that each $E_{n}subset Bsetminus F$ but $cup_{n}E_{n}=B$?
real-analysis measure-theory elementary-set-theory
real-analysis measure-theory elementary-set-theory
asked Nov 19 at 5:29
M. Rahmat
291211
asked Nov 19 at 5:29
M. Rahmat
291211
asked Nov 19 at 5:29
M. Rahmat
291211
asked Nov 19 at 5:29
M. Rahmat
291211
asked Nov 19 at 5:29
M. Rahmat
291211
291211
closed as unclear what you're asking by Kavi Rama Murthy, user90369, user10354138, Shailesh, Cesareo Nov 20 at 1:15
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
closed as unclear what you're asking by Kavi Rama Murthy, user90369, user10354138, Shailesh, Cesareo Nov 20 at 1:15
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
4
Not if $Bcap F$ is nonempty. Is this actually the question you wished to ask?
– Lord Shark the Unknown
Nov 19 at 5:32
$F$ is a part of $B$ and so their intersection is not empty.
– M. Rahmat
Nov 19 at 6:16
add a comment |
4
Not if $Bcap F$ is nonempty. Is this actually the question you wished to ask?
– Lord Shark the Unknown
Nov 19 at 5:32
$F$ is a part of $B$ and so their intersection is not empty.
– M. Rahmat
Nov 19 at 6:16
4
4
Not if $Bcap F$ is nonempty. Is this actually the question you wished to ask?
– Lord Shark the Unknown
Nov 19 at 5:32
Not if $Bcap F$ is nonempty. Is this actually the question you wished to ask?
– Lord Shark the Unknown
Nov 19 at 5:32
$F$ is a part of $B$ and so their intersection is not empty.
– M. Rahmat
Nov 19 at 6:16
$F$ is a part of $B$ and so their intersection is not empty.
– M. Rahmat
Nov 19 at 6:16
add a comment |
1 Answer
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No.
In general if $A_alpha$ are subsets of a set $D$, then $bigcup_{alpha} A_alpha$ is also a subset of $D$, since any element of the union is an element of some $A_alpha$ and hence an element of $D$.
Thus as Lord Shark the Unknown points out in the comments, if $Bcap F$ is nonempty then $Bsetminus F$ is a proper subset of $B$, so since each $E_n$ is a subset of $Bsetminus F$, we have that $bigcup_n E_n subseteq Bsetminus F subsetneq B$.
answered Nov 20 at 0:53
jgon
11.1k11839
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
No.
In general if $A_alpha$ are subsets of a set $D$, then $bigcup_{alpha} A_alpha$ is also a subset of $D$, since any element of the union is an element of some $A_alpha$ and hence an element of $D$.
Thus as Lord Shark the Unknown points out in the comments, if $Bcap F$ is nonempty then $Bsetminus F$ is a proper subset of $B$, so since each $E_n$ is a subset of $Bsetminus F$, we have that $bigcup_n E_n subseteq Bsetminus F subsetneq B$.
answered Nov 20 at 0:53
jgon
11.1k11839
add a comment |
up vote
0
down vote
accepted
No.
In general if $A_alpha$ are subsets of a set $D$, then $bigcup_{alpha} A_alpha$ is also a subset of $D$, since any element of the union is an element of some $A_alpha$ and hence an element of $D$.
Thus as Lord Shark the Unknown points out in the comments, if $Bcap F$ is nonempty then $Bsetminus F$ is a proper subset of $B$, so since each $E_n$ is a subset of $Bsetminus F$, we have that $bigcup_n E_n subseteq Bsetminus F subsetneq B$.
answered Nov 20 at 0:53
jgon
11.1k11839
add a comment |
up vote
0
down vote
accepted
up vote
0
down vote
accepted
No.
In general if $A_alpha$ are subsets of a set $D$, then $bigcup_{alpha} A_alpha$ is also a subset of $D$, since any element of the union is an element of some $A_alpha$ and hence an element of $D$.
Thus as Lord Shark the Unknown points out in the comments, if $Bcap F$ is nonempty then $Bsetminus F$ is a proper subset of $B$, so since each $E_n$ is a subset of $Bsetminus F$, we have that $bigcup_n E_n subseteq Bsetminus F subsetneq B$.
answered Nov 20 at 0:53
jgon
11.1k11839
No.
In general if $A_alpha$ are subsets of a set $D$, then $bigcup_{alpha} A_alpha$ is also a subset of $D$, since any element of the union is an element of some $A_alpha$ and hence an element of $D$.
Thus as Lord Shark the Unknown points out in the comments, if $Bcap F$ is nonempty then $Bsetminus F$ is a proper subset of $B$, so since each $E_n$ is a subset of $Bsetminus F$, we have that $bigcup_n E_n subseteq Bsetminus F subsetneq B$.
answered Nov 20 at 0:53
jgon
11.1k11839
answered Nov 20 at 0:53
jgon
11.1k11839
answered Nov 20 at 0:53
jgon
11.1k11839
answered Nov 20 at 0:53
jgon
11.1k11839
11.1k11839
add a comment |
add a comment |
4
Not if $Bcap F$ is nonempty. Is this actually the question you wished to ask?
– Lord Shark the Unknown
Nov 19 at 5:32
$F$ is a part of $B$ and so their intersection is not empty.
– M. Rahmat
Nov 19 at 6:16