Question on point set topology
1
$begingroup$
Does there exist a closed set which is an intersection of a collection of infinite open sets?
analysis
asked 1 hour ago
Tony TongTony Tong
322
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Tony Tong is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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$endgroup$
add a comment |
1
$begingroup$
Does there exist a closed set which is an intersection of a collection of infinite open sets?
analysis
asked 1 hour ago
Tony TongTony Tong
322
New contributor
Tony Tong is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
5
$begingroup$
Intersect $(-tfrac{1}{n}, tfrac{1}{n})$ for $n = 1, 2, ldots$ and consider what set you get
$endgroup$
– Brevan Ellefsen
1 hour ago
$begingroup$
Oh it will get ${0}$
$endgroup$
– Tony Tong
59 mins ago
add a comment |
1
1
1
$begingroup$
Does there exist a closed set which is an intersection of a collection of infinite open sets?
analysis
asked 1 hour ago
Tony TongTony Tong
322
New contributor
Tony Tong is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
Does there exist a closed set which is an intersection of a collection of infinite open sets?
analysis
analysis
asked 1 hour ago
Tony TongTony Tong
322
New contributor
Tony Tong is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 1 hour ago
Tony TongTony Tong
322
New contributor
Tony Tong is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 1 hour ago
Tony TongTony Tong
322
New contributor
Tony Tong is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 1 hour ago
Tony TongTony Tong
322
asked 1 hour ago
Tony TongTony Tong
322
322
New contributor
Tony Tong is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Tony Tong is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Tony Tong is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
5
$begingroup$
Intersect $(-tfrac{1}{n}, tfrac{1}{n})$ for $n = 1, 2, ldots$ and consider what set you get
$endgroup$
– Brevan Ellefsen
1 hour ago
$begingroup$
Oh it will get ${0}$
$endgroup$
– Tony Tong
59 mins ago
add a comment |
5
$begingroup$
Intersect $(-tfrac{1}{n}, tfrac{1}{n})$ for $n = 1, 2, ldots$ and consider what set you get
$endgroup$
– Brevan Ellefsen
1 hour ago
$begingroup$
Oh it will get ${0}$
$endgroup$
– Tony Tong
59 mins ago
5
5
$begingroup$
Intersect $(-tfrac{1}{n}, tfrac{1}{n})$ for $n = 1, 2, ldots$ and consider what set you get
$endgroup$
– Brevan Ellefsen
1 hour ago
$begingroup$
Intersect $(-tfrac{1}{n}, tfrac{1}{n})$ for $n = 1, 2, ldots$ and consider what set you get
$endgroup$
– Brevan Ellefsen
1 hour ago
$begingroup$
Oh it will get ${0}$
$endgroup$
– Tony Tong
59 mins ago
$begingroup$
Oh it will get ${0}$
$endgroup$
– Tony Tong
59 mins ago
add a comment |
1 Answer
1
active
oldest
votes
6
$begingroup$
$$mathbb{R}capmathbb{R}capmathbb{R}capcdots$$
answered 1 hour ago
parsiadparsiad
18.5k32453
$endgroup$
$begingroup$
But R is an open set, the intersection is also R so it is still an open set
$endgroup$
– Tony Tong
1 hour ago
1
$begingroup$
And also closed
$endgroup$
– Keen-ameteur
1 hour ago
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
57 mins ago
$begingroup$
@BrevanEllefsen: +1, but I couldn't resist... :-)
$endgroup$
– parsiad
57 mins ago
$begingroup$
(+1) That's a bad answer from the pedagogical point of view… but it's really funny too!
$endgroup$
– José Carlos Santos
41 secs ago
add a comment |
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Tony Tong is a new contributor. Be nice, and check out our Code of Conduct.
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
6
$begingroup$
$$mathbb{R}capmathbb{R}capmathbb{R}capcdots$$
answered 1 hour ago
parsiadparsiad
18.5k32453
$endgroup$
$begingroup$
But R is an open set, the intersection is also R so it is still an open set
$endgroup$
– Tony Tong
1 hour ago
1
$begingroup$
And also closed
$endgroup$
– Keen-ameteur
1 hour ago
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
57 mins ago
$begingroup$
@BrevanEllefsen: +1, but I couldn't resist... :-)
$endgroup$
– parsiad
57 mins ago
$begingroup$
(+1) That's a bad answer from the pedagogical point of view… but it's really funny too!
$endgroup$
– José Carlos Santos
41 secs ago
add a comment |
6
$begingroup$
$$mathbb{R}capmathbb{R}capmathbb{R}capcdots$$
answered 1 hour ago
parsiadparsiad
18.5k32453
$endgroup$
$begingroup$
But R is an open set, the intersection is also R so it is still an open set
$endgroup$
– Tony Tong
1 hour ago
1
$begingroup$
And also closed
$endgroup$
– Keen-ameteur
1 hour ago
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
57 mins ago
$begingroup$
@BrevanEllefsen: +1, but I couldn't resist... :-)
$endgroup$
– parsiad
57 mins ago
$begingroup$
(+1) That's a bad answer from the pedagogical point of view… but it's really funny too!
$endgroup$
– José Carlos Santos
41 secs ago
add a comment |
6
6
6
$begingroup$
$$mathbb{R}capmathbb{R}capmathbb{R}capcdots$$
answered 1 hour ago
parsiadparsiad
18.5k32453
$endgroup$
$$mathbb{R}capmathbb{R}capmathbb{R}capcdots$$
answered 1 hour ago
parsiadparsiad
18.5k32453
answered 1 hour ago
parsiadparsiad
18.5k32453
answered 1 hour ago
parsiadparsiad
18.5k32453
answered 1 hour ago
parsiadparsiad
18.5k32453
18.5k32453
$begingroup$
But R is an open set, the intersection is also R so it is still an open set
$endgroup$
– Tony Tong
1 hour ago
1
$begingroup$
And also closed
$endgroup$
– Keen-ameteur
1 hour ago
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
57 mins ago
$begingroup$
@BrevanEllefsen: +1, but I couldn't resist... :-)
$endgroup$
– parsiad
57 mins ago
$begingroup$
(+1) That's a bad answer from the pedagogical point of view… but it's really funny too!
$endgroup$
– José Carlos Santos
41 secs ago
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
1 hour ago
1
$begingroup$
And also closed
$endgroup$
– Keen-ameteur
1 hour ago
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
57 mins ago
$begingroup$
@BrevanEllefsen: +1, but I couldn't resist... :-)
$endgroup$
– parsiad
57 mins ago
$begingroup$
(+1) That's a bad answer from the pedagogical point of view… but it's really funny too!
$endgroup$
– José Carlos Santos
41 secs ago
$begingroup$
But R is an open set, the intersection is also R so it is still an open set
$endgroup$
– Tony Tong
1 hour ago
$begingroup$
But R is an open set, the intersection is also R so it is still an open set
$endgroup$
– Tony Tong
1 hour ago
1
1
$begingroup$
And also closed
$endgroup$
– Keen-ameteur
1 hour ago
$begingroup$
And also closed
$endgroup$
– Keen-ameteur
1 hour ago
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
57 mins ago
$begingroup$
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
57 mins ago
$begingroup$
@BrevanEllefsen: +1, but I couldn't resist... :-)
$endgroup$
– parsiad
57 mins ago
$begingroup$
@BrevanEllefsen: +1, but I couldn't resist... :-)
$endgroup$
– parsiad
57 mins ago
$begingroup$
(+1) That's a bad answer from the pedagogical point of view… but it's really funny too!
$endgroup$
– José Carlos Santos
41 secs ago
$begingroup$
(+1) That's a bad answer from the pedagogical point of view… but it's really funny too!
$endgroup$
– José Carlos Santos
41 secs ago
add a comment |
Tony Tong is a new contributor. Be nice, and check out our Code of Conduct.
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Tony Tong is a new contributor. Be nice, and check out our Code of Conduct.
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5
$begingroup$
Intersect $(-tfrac{1}{n}, tfrac{1}{n})$ for $n = 1, 2, ldots$ and consider what set you get
$endgroup$
– Brevan Ellefsen
1 hour ago
$begingroup$
Oh it will get ${0}$
$endgroup$
– Tony Tong
59 mins ago