Does there exist a closed set which is an intersection of a collection of infinite open sets? [closed]
Clash Royale CLAN TAG#URR8PPP
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Does there exist a closed set which is an intersection of a collection of infinite open sets?
analysis
edited Mar 18 at 8:39
YuiTo Cheng
2,43841037
asked Mar 18 at 5:50
Tony TongTony Tong
364
$endgroup$
closed as off-topic by Eevee Trainer, Shalop, Shailesh, user21820, José Carlos Santos Mar 18 at 10:40
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." – Eevee Trainer, Shalop, Shailesh, user21820, José Carlos Santos
add a comment |
$begingroup$
Does there exist a closed set which is an intersection of a collection of infinite open sets?
analysis
edited Mar 18 at 8:39
YuiTo Cheng
2,43841037
asked Mar 18 at 5:50
Tony TongTony Tong
364
$endgroup$
closed as off-topic by Eevee Trainer, Shalop, Shailesh, user21820, José Carlos Santos Mar 18 at 10:40
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." – Eevee Trainer, Shalop, Shailesh, user21820, José Carlos Santos
8
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Intersect $(-tfrac1n, tfrac1n)$ for $n = 1, 2, ldots$ and consider what set you get
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– Brevan Ellefsen
Mar 18 at 5:52
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Oh it will get $0$
$endgroup$
– Tony Tong
Mar 18 at 5:57
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@BrevanEllefsen: The question only asks for a "collection of infinite open sets", so just the singleton $(0,1)$ suffices. But Tony, please read how to ask a good question.
$endgroup$
– user21820
Mar 18 at 9:23
add a comment |
$begingroup$
Does there exist a closed set which is an intersection of a collection of infinite open sets?
analysis
edited Mar 18 at 8:39
YuiTo Cheng
2,43841037
asked Mar 18 at 5:50
Tony TongTony Tong
364
$endgroup$
Does there exist a closed set which is an intersection of a collection of infinite open sets?
analysis
analysis
edited Mar 18 at 8:39
YuiTo Cheng
2,43841037
asked Mar 18 at 5:50
Tony TongTony Tong
364
edited Mar 18 at 8:39
YuiTo Cheng
2,43841037
asked Mar 18 at 5:50
Tony TongTony Tong
364
edited Mar 18 at 8:39
YuiTo Cheng
2,43841037
edited Mar 18 at 8:39
YuiTo Cheng
2,43841037
edited Mar 18 at 8:39
YuiTo Cheng
2,43841037
2,43841037
asked Mar 18 at 5:50
Tony TongTony Tong
364
asked Mar 18 at 5:50
Tony TongTony Tong
364
asked Mar 18 at 5:50
Tony TongTony Tong
364
364
closed as off-topic by Eevee Trainer, Shalop, Shailesh, user21820, José Carlos Santos Mar 18 at 10:40
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." – Eevee Trainer, Shalop, Shailesh, user21820, José Carlos Santos
closed as off-topic by Eevee Trainer, Shalop, Shailesh, user21820, José Carlos Santos Mar 18 at 10:40
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." – Eevee Trainer, Shalop, Shailesh, user21820, José Carlos Santos
8
$begingroup$
Intersect $(-tfrac1n, tfrac1n)$ for $n = 1, 2, ldots$ and consider what set you get
$endgroup$
– Brevan Ellefsen
Mar 18 at 5:52
$begingroup$
Oh it will get $0$
$endgroup$
– Tony Tong
Mar 18 at 5:57
$begingroup$
@BrevanEllefsen: The question only asks for a "collection of infinite open sets", so just the singleton $(0,1)$ suffices. But Tony, please read how to ask a good question.
$endgroup$
– user21820
Mar 18 at 9:23
add a comment |
8
$begingroup$
Intersect $(-tfrac1n, tfrac1n)$ for $n = 1, 2, ldots$ and consider what set you get
$endgroup$
– Brevan Ellefsen
Mar 18 at 5:52
$begingroup$
Oh it will get $0$
$endgroup$
– Tony Tong
Mar 18 at 5:57
$begingroup$
@BrevanEllefsen: The question only asks for a "collection of infinite open sets", so just the singleton $(0,1)$ suffices. But Tony, please read how to ask a good question.
$endgroup$
– user21820
Mar 18 at 9:23
8
8
$begingroup$
Intersect $(-tfrac1n, tfrac1n)$ for $n = 1, 2, ldots$ and consider what set you get
$endgroup$
– Brevan Ellefsen
Mar 18 at 5:52
$begingroup$
Intersect $(-tfrac1n, tfrac1n)$ for $n = 1, 2, ldots$ and consider what set you get
$endgroup$
– Brevan Ellefsen
Mar 18 at 5:52
$begingroup$
Oh it will get $0$
$endgroup$
– Tony Tong
Mar 18 at 5:57
$begingroup$
Oh it will get $0$
$endgroup$
– Tony Tong
Mar 18 at 5:57
$begingroup$
@BrevanEllefsen: The question only asks for a "collection of infinite open sets", so just the singleton $(0,1)$ suffices. But Tony, please read how to ask a good question.
$endgroup$
– user21820
Mar 18 at 9:23
$begingroup$
@BrevanEllefsen: The question only asks for a "collection of infinite open sets", so just the singleton $(0,1)$ suffices. But Tony, please read how to ask a good question.
$endgroup$
– user21820
Mar 18 at 9:23
add a comment |
2 Answers
2
active
oldest
votes
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$$mathbbRcapmathbbRcapmathbbRcapcdots$$
answered Mar 18 at 5:53
parsiadparsiad
18.8k32554
$endgroup$
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But R is an open set, the intersection is also R so it is still an open set
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– Tony Tong
Mar 18 at 5:54
2
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And also closed
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– Keen-ameteur
Mar 18 at 5:55
1
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While extremely simple, this example has the unfortunate side effect of also being open, which could further confound the OP who seems to be wondering why the intersection of open sets is not open in general (admittedly, the OP is probably also drawing a false dichotomy between open and closed sets, so I suppose this helps with that)
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– Brevan Ellefsen
Mar 18 at 5:59
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@BrevanEllefsen: +1, but I couldn't resist... :-)
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– parsiad
Mar 18 at 6:00
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(+1) That's a bad answer from the pedagogical point of view… but it's really funny too!
$endgroup$
– José Carlos Santos
Mar 18 at 6:56
add a comment |
$begingroup$
As other answers and comments point out, this does indeed hold. In fact, every closed set in $mathbb R$ is a (countable) intersection of open sets. This holds in any metric space $(X,d)$. Here is a proof in the general metric case:
Suppose $Asubset X$ is closed. For each $n$, let
$$
U_n :=bigcup_ain A B(a,1/n).
$$
Note that $U_n$ is open (union of open sets). We claim that
$$
A = bigcap_ninmathbb N U_n.
$$
First note that $Asubset bigcap U_n$ since $Asubset U_n$ for each $n$ (by construction!). Also, if $xinbigcap U_n$, then for each $n$ we have some $a_n$ so that $xin B(a_n, 1/n)$, i.e. $d(x,a_n) < 1/n$ for each $n$. This implies that $a_nrightarrow x$ as $nrightarrowinfty$. Since $A$ is closed, it follows that $xin A$. QED.
answered Mar 18 at 6:59
o.h.o.h.
6917
$endgroup$
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
$$mathbbRcapmathbbRcapmathbbRcapcdots$$
answered Mar 18 at 5:53
parsiadparsiad
18.8k32554
$endgroup$
$begingroup$
But R is an open set, the intersection is also R so it is still an open set
$endgroup$
– Tony Tong
Mar 18 at 5:54
2
$begingroup$
And also closed
$endgroup$
– Keen-ameteur
Mar 18 at 5:55
1
$begingroup$
While extremely simple, this example has the unfortunate side effect of also being open, which could further confound the OP who seems to be wondering why the intersection of open sets is not open in general (admittedly, the OP is probably also drawing a false dichotomy between open and closed sets, so I suppose this helps with that)
$endgroup$
– Brevan Ellefsen
Mar 18 at 5:59
$begingroup$
@BrevanEllefsen: +1, but I couldn't resist... :-)
$endgroup$
– parsiad
Mar 18 at 6:00
$begingroup$
(+1) That's a bad answer from the pedagogical point of view… but it's really funny too!
$endgroup$
– José Carlos Santos
Mar 18 at 6:56
add a comment |
$begingroup$
$$mathbbRcapmathbbRcapmathbbRcapcdots$$
answered Mar 18 at 5:53
parsiadparsiad
18.8k32554
$endgroup$
$begingroup$
But R is an open set, the intersection is also R so it is still an open set
$endgroup$
– Tony Tong
Mar 18 at 5:54
2
$begingroup$
And also closed
$endgroup$
– Keen-ameteur
Mar 18 at 5:55
1
$begingroup$
While extremely simple, this example has the unfortunate side effect of also being open, which could further confound the OP who seems to be wondering why the intersection of open sets is not open in general (admittedly, the OP is probably also drawing a false dichotomy between open and closed sets, so I suppose this helps with that)
$endgroup$
– Brevan Ellefsen
Mar 18 at 5:59
$begingroup$
@BrevanEllefsen: +1, but I couldn't resist... :-)
$endgroup$
– parsiad
Mar 18 at 6:00
$begingroup$
(+1) That's a bad answer from the pedagogical point of view… but it's really funny too!
$endgroup$
– José Carlos Santos
Mar 18 at 6:56
add a comment |
$begingroup$
$$mathbbRcapmathbbRcapmathbbRcapcdots$$
answered Mar 18 at 5:53
parsiadparsiad
18.8k32554
$endgroup$
$$mathbbRcapmathbbRcapmathbbRcapcdots$$
answered Mar 18 at 5:53
parsiadparsiad
18.8k32554
answered Mar 18 at 5:53
parsiadparsiad
18.8k32554
answered Mar 18 at 5:53
parsiadparsiad
18.8k32554
answered Mar 18 at 5:53
parsiadparsiad
18.8k32554
18.8k32554
$begingroup$
But R is an open set, the intersection is also R so it is still an open set
$endgroup$
– Tony Tong
Mar 18 at 5:54
2
$begingroup$
And also closed
$endgroup$
– Keen-ameteur
Mar 18 at 5:55
1
$begingroup$
While extremely simple, this example has the unfortunate side effect of also being open, which could further confound the OP who seems to be wondering why the intersection of open sets is not open in general (admittedly, the OP is probably also drawing a false dichotomy between open and closed sets, so I suppose this helps with that)
$endgroup$
– Brevan Ellefsen
Mar 18 at 5:59
$begingroup$
@BrevanEllefsen: +1, but I couldn't resist... :-)
$endgroup$
– parsiad
Mar 18 at 6:00
$begingroup$
(+1) That's a bad answer from the pedagogical point of view… but it's really funny too!
$endgroup$
– José Carlos Santos
Mar 18 at 6:56
add a comment |
$begingroup$
But R is an open set, the intersection is also R so it is still an open set
$endgroup$
– Tony Tong
Mar 18 at 5:54
2
$begingroup$
And also closed
$endgroup$
– Keen-ameteur
Mar 18 at 5:55
1
$begingroup$
While extremely simple, this example has the unfortunate side effect of also being open, which could further confound the OP who seems to be wondering why the intersection of open sets is not open in general (admittedly, the OP is probably also drawing a false dichotomy between open and closed sets, so I suppose this helps with that)
$endgroup$
– Brevan Ellefsen
Mar 18 at 5:59
$begingroup$
@BrevanEllefsen: +1, but I couldn't resist... :-)
$endgroup$
– parsiad
Mar 18 at 6:00
$begingroup$
(+1) That's a bad answer from the pedagogical point of view… but it's really funny too!
$endgroup$
– José Carlos Santos
Mar 18 at 6:56
$begingroup$
But R is an open set, the intersection is also R so it is still an open set
$endgroup$
– Tony Tong
Mar 18 at 5:54
$begingroup$
But R is an open set, the intersection is also R so it is still an open set
$endgroup$
– Tony Tong
Mar 18 at 5:54
2
2
$begingroup$
And also closed
$endgroup$
– Keen-ameteur
Mar 18 at 5:55
$begingroup$
And also closed
$endgroup$
– Keen-ameteur
Mar 18 at 5:55
1
1
$begingroup$
While extremely simple, this example has the unfortunate side effect of also being open, which could further confound the OP who seems to be wondering why the intersection of open sets is not open in general (admittedly, the OP is probably also drawing a false dichotomy between open and closed sets, so I suppose this helps with that)
$endgroup$
– Brevan Ellefsen
Mar 18 at 5:59
$begingroup$
While extremely simple, this example has the unfortunate side effect of also being open, which could further confound the OP who seems to be wondering why the intersection of open sets is not open in general (admittedly, the OP is probably also drawing a false dichotomy between open and closed sets, so I suppose this helps with that)
$endgroup$
– Brevan Ellefsen
Mar 18 at 5:59
$begingroup$
@BrevanEllefsen: +1, but I couldn't resist... :-)
$endgroup$
– parsiad
Mar 18 at 6:00
$begingroup$
@BrevanEllefsen: +1, but I couldn't resist... :-)
$endgroup$
– parsiad
Mar 18 at 6:00
$begingroup$
(+1) That's a bad answer from the pedagogical point of view… but it's really funny too!
$endgroup$
– José Carlos Santos
Mar 18 at 6:56
$begingroup$
(+1) That's a bad answer from the pedagogical point of view… but it's really funny too!
$endgroup$
– José Carlos Santos
Mar 18 at 6:56
add a comment |
$begingroup$
As other answers and comments point out, this does indeed hold. In fact, every closed set in $mathbb R$ is a (countable) intersection of open sets. This holds in any metric space $(X,d)$. Here is a proof in the general metric case:
Suppose $Asubset X$ is closed. For each $n$, let
$$
U_n :=bigcup_ain A B(a,1/n).
$$
Note that $U_n$ is open (union of open sets). We claim that
$$
A = bigcap_ninmathbb N U_n.
$$
First note that $Asubset bigcap U_n$ since $Asubset U_n$ for each $n$ (by construction!). Also, if $xinbigcap U_n$, then for each $n$ we have some $a_n$ so that $xin B(a_n, 1/n)$, i.e. $d(x,a_n) < 1/n$ for each $n$. This implies that $a_nrightarrow x$ as $nrightarrowinfty$. Since $A$ is closed, it follows that $xin A$. QED.
answered Mar 18 at 6:59
o.h.o.h.
6917
$endgroup$
add a comment |
$begingroup$
As other answers and comments point out, this does indeed hold. In fact, every closed set in $mathbb R$ is a (countable) intersection of open sets. This holds in any metric space $(X,d)$. Here is a proof in the general metric case:
Suppose $Asubset X$ is closed. For each $n$, let
$$
U_n :=bigcup_ain A B(a,1/n).
$$
Note that $U_n$ is open (union of open sets). We claim that
$$
A = bigcap_ninmathbb N U_n.
$$
First note that $Asubset bigcap U_n$ since $Asubset U_n$ for each $n$ (by construction!). Also, if $xinbigcap U_n$, then for each $n$ we have some $a_n$ so that $xin B(a_n, 1/n)$, i.e. $d(x,a_n) < 1/n$ for each $n$. This implies that $a_nrightarrow x$ as $nrightarrowinfty$. Since $A$ is closed, it follows that $xin A$. QED.
answered Mar 18 at 6:59
o.h.o.h.
6917
$endgroup$
add a comment |
$begingroup$
As other answers and comments point out, this does indeed hold. In fact, every closed set in $mathbb R$ is a (countable) intersection of open sets. This holds in any metric space $(X,d)$. Here is a proof in the general metric case:
Suppose $Asubset X$ is closed. For each $n$, let
$$
U_n :=bigcup_ain A B(a,1/n).
$$
Note that $U_n$ is open (union of open sets). We claim that
$$
A = bigcap_ninmathbb N U_n.
$$
First note that $Asubset bigcap U_n$ since $Asubset U_n$ for each $n$ (by construction!). Also, if $xinbigcap U_n$, then for each $n$ we have some $a_n$ so that $xin B(a_n, 1/n)$, i.e. $d(x,a_n) < 1/n$ for each $n$. This implies that $a_nrightarrow x$ as $nrightarrowinfty$. Since $A$ is closed, it follows that $xin A$. QED.
answered Mar 18 at 6:59
o.h.o.h.
6917
$endgroup$
As other answers and comments point out, this does indeed hold. In fact, every closed set in $mathbb R$ is a (countable) intersection of open sets. This holds in any metric space $(X,d)$. Here is a proof in the general metric case:
Suppose $Asubset X$ is closed. For each $n$, let
$$
U_n :=bigcup_ain A B(a,1/n).
$$
Note that $U_n$ is open (union of open sets). We claim that
$$
A = bigcap_ninmathbb N U_n.
$$
First note that $Asubset bigcap U_n$ since $Asubset U_n$ for each $n$ (by construction!). Also, if $xinbigcap U_n$, then for each $n$ we have some $a_n$ so that $xin B(a_n, 1/n)$, i.e. $d(x,a_n) < 1/n$ for each $n$. This implies that $a_nrightarrow x$ as $nrightarrowinfty$. Since $A$ is closed, it follows that $xin A$. QED.
answered Mar 18 at 6:59
o.h.o.h.
6917
edited Mar 18 at 7:04
answered Mar 18 at 6:59
o.h.o.h.
6917
answered Mar 18 at 6:59
o.h.o.h.
6917
answered Mar 18 at 6:59
o.h.o.h.
6917
6917
add a comment |
add a comment |
8
$begingroup$
Intersect $(-tfrac1n, tfrac1n)$ for $n = 1, 2, ldots$ and consider what set you get
$endgroup$
– Brevan Ellefsen
Mar 18 at 5:52
$begingroup$
Oh it will get $0$
$endgroup$
– Tony Tong
Mar 18 at 5:57
$begingroup$
@BrevanEllefsen: The question only asks for a "collection of infinite open sets", so just the singleton $(0,1)$ suffices. But Tony, please read how to ask a good question.
$endgroup$
– user21820
Mar 18 at 9:23