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 33 mins ago
Tony TongTony Tong
292
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Tony Tong is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
1
$begingroup$
Does there exist a closed set which is an intersection of a collection of infinite open sets?
analysis
asked 33 mins ago
Tony TongTony Tong
292
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
30 mins ago
$begingroup$
Oh it will get ${0}$
$endgroup$
– Tony Tong
25 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 33 mins ago
Tony TongTony Tong
292
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 33 mins ago
Tony TongTony Tong
292
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 33 mins ago
Tony TongTony Tong
292
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 33 mins ago
Tony TongTony Tong
292
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 33 mins ago
Tony TongTony Tong
292
asked 33 mins ago
Tony TongTony Tong
292
292
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
30 mins ago
$begingroup$
Oh it will get ${0}$
$endgroup$
– Tony Tong
25 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
30 mins ago
$begingroup$
Oh it will get ${0}$
$endgroup$
– Tony Tong
25 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
30 mins ago
$begingroup$
Intersect $(-tfrac{1}{n}, tfrac{1}{n})$ for $n = 1, 2, ldots$ and consider what set you get
$endgroup$
– Brevan Ellefsen
30 mins ago
$begingroup$
Oh it will get ${0}$
$endgroup$
– Tony Tong
25 mins ago
$begingroup$
Oh it will get ${0}$
$endgroup$
– Tony Tong
25 mins ago
add a comment |
1 Answer
1
active
oldest
votes
4
$begingroup$
$$mathbb{R}capmathbb{R}capmathbb{R}capcdots$$
answered 30 mins ago
parsiadparsiad
18.4k32453
$endgroup$
$begingroup$
But R is an open set, the intersection is also R so it is still an open set
$endgroup$
– Tony Tong
29 mins ago
1
$begingroup$
And also closed
$endgroup$
– Keen-ameteur
27 mins 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
23 mins ago
$begingroup$
@BrevanEllefsen: +1, but I couldn't resist... :-)
$endgroup$
– parsiad
23 mins 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
4
$begingroup$
$$mathbb{R}capmathbb{R}capmathbb{R}capcdots$$
answered 30 mins ago
parsiadparsiad
18.4k32453
$endgroup$
$begingroup$
But R is an open set, the intersection is also R so it is still an open set
$endgroup$
– Tony Tong
29 mins ago
1
$begingroup$
And also closed
$endgroup$
– Keen-ameteur
27 mins 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
23 mins ago
$begingroup$
@BrevanEllefsen: +1, but I couldn't resist... :-)
$endgroup$
– parsiad
23 mins ago
add a comment |
4
$begingroup$
$$mathbb{R}capmathbb{R}capmathbb{R}capcdots$$
answered 30 mins ago
parsiadparsiad
18.4k32453
$endgroup$
$begingroup$
But R is an open set, the intersection is also R so it is still an open set
$endgroup$
– Tony Tong
29 mins ago
1
$begingroup$
And also closed
$endgroup$
– Keen-ameteur
27 mins 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
23 mins ago
$begingroup$
@BrevanEllefsen: +1, but I couldn't resist... :-)
$endgroup$
– parsiad
23 mins ago
add a comment |
4
4
4
$begingroup$
$$mathbb{R}capmathbb{R}capmathbb{R}capcdots$$
answered 30 mins ago
parsiadparsiad
18.4k32453
$endgroup$
$$mathbb{R}capmathbb{R}capmathbb{R}capcdots$$
answered 30 mins ago
parsiadparsiad
18.4k32453
answered 30 mins ago
parsiadparsiad
18.4k32453
answered 30 mins ago
parsiadparsiad
18.4k32453
answered 30 mins ago
parsiadparsiad
18.4k32453
18.4k32453
$begingroup$
But R is an open set, the intersection is also R so it is still an open set
$endgroup$
– Tony Tong
29 mins ago
1
$begingroup$
And also closed
$endgroup$
– Keen-ameteur
27 mins 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
23 mins ago
$begingroup$
@BrevanEllefsen: +1, but I couldn't resist... :-)
$endgroup$
– parsiad
23 mins 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
29 mins ago
1
$begingroup$
And also closed
$endgroup$
– Keen-ameteur
27 mins 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
23 mins ago
$begingroup$
@BrevanEllefsen: +1, but I couldn't resist... :-)
$endgroup$
– parsiad
23 mins ago
$begingroup$
But R is an open set, the intersection is also R so it is still an open set
$endgroup$
– Tony Tong
29 mins ago
$begingroup$
But R is an open set, the intersection is also R so it is still an open set
$endgroup$
– Tony Tong
29 mins ago
1
1
$begingroup$
And also closed
$endgroup$
– Keen-ameteur
27 mins ago
$begingroup$
And also closed
$endgroup$
– Keen-ameteur
27 mins 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
23 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
23 mins ago
$begingroup$
@BrevanEllefsen: +1, but I couldn't resist... :-)
$endgroup$
– parsiad
23 mins ago
$begingroup$
@BrevanEllefsen: +1, but I couldn't resist... :-)
$endgroup$
– parsiad
23 mins 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.
Tony Tong is a new contributor. Be nice, and check out our Code of Conduct.
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
30 mins ago
$begingroup$
Oh it will get ${0}$
$endgroup$
– Tony Tong
25 mins ago