Question on point set topologyDefinition of Borel setsA “complementary” topologyFinite vs infinite...
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Question on point set topology
Definition of Borel setsA “complementary” topologyFinite vs infinite distinction in Rudin's AnalysisThe set of rationals in $(0,1)$ is not a $G_delta$Limit point of an infinite subset of a compact setIf $U ⊂ mathbb{R}^n$ is open and $B ⊂ U$, then why is it that $B$ relatively open in $U$ if and only if $B$ is open?Question about Theorem 2.24 in Baby RudinShowing that if closed subsets don't intersect then there exists open sets in which they exist that also don't intersectDifference between closure and closed cover of a setIs there an analogue for a compact set using closed sets?
$begingroup$
Does there exist a closed set which is an intersection of a collection of infinite open sets?
analysis
asked 46 mins ago
Tony TongTony Tong
272
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.
$endgroup$
add a comment |
$begingroup$
Does there exist a closed set which is an intersection of a collection of infinite open sets?
analysis
asked 46 mins ago
Tony TongTony Tong
272
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
44 mins ago
$begingroup$
Oh it will get ${0}$
$endgroup$
– Tony Tong
39 mins ago
add a comment |
$begingroup$
Does there exist a closed set which is an intersection of a collection of infinite open sets?
analysis
asked 46 mins ago
Tony TongTony Tong
272
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 46 mins ago
Tony TongTony Tong
272
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 46 mins ago
Tony TongTony Tong
272
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 46 mins ago
Tony TongTony Tong
272
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 46 mins ago
Tony TongTony Tong
272
asked 46 mins ago
Tony TongTony Tong
272
272
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
44 mins ago
$begingroup$
Oh it will get ${0}$
$endgroup$
– Tony Tong
39 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
44 mins ago
$begingroup$
Oh it will get ${0}$
$endgroup$
– Tony Tong
39 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
44 mins ago
$begingroup$
Intersect $(-tfrac{1}{n}, tfrac{1}{n})$ for $n = 1, 2, ldots$ and consider what set you get
$endgroup$
– Brevan Ellefsen
44 mins ago
$begingroup$
Oh it will get ${0}$
$endgroup$
– Tony Tong
39 mins ago
$begingroup$
Oh it will get ${0}$
$endgroup$
– Tony Tong
39 mins ago
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
$$mathbb{R}capmathbb{R}capmathbb{R}capcdots$$
answered 44 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
42 mins ago
1
$begingroup$
And also closed
$endgroup$
– Keen-ameteur
41 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
37 mins ago
$begingroup$
@BrevanEllefsen: +1, but I couldn't resist... :-)
$endgroup$
– parsiad
36 mins ago
add a comment |
Your Answer
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
$$mathbb{R}capmathbb{R}capmathbb{R}capcdots$$
answered 44 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
42 mins ago
1
$begingroup$
And also closed
$endgroup$
– Keen-ameteur
41 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
37 mins ago
$begingroup$
@BrevanEllefsen: +1, but I couldn't resist... :-)
$endgroup$
– parsiad
36 mins ago
add a comment |
$begingroup$
$$mathbb{R}capmathbb{R}capmathbb{R}capcdots$$
answered 44 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
42 mins ago
1
$begingroup$
And also closed
$endgroup$
– Keen-ameteur
41 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
37 mins ago
$begingroup$
@BrevanEllefsen: +1, but I couldn't resist... :-)
$endgroup$
– parsiad
36 mins ago
add a comment |
$begingroup$
$$mathbb{R}capmathbb{R}capmathbb{R}capcdots$$
answered 44 mins ago
parsiadparsiad
18.4k32453
$endgroup$
$$mathbb{R}capmathbb{R}capmathbb{R}capcdots$$
answered 44 mins ago
parsiadparsiad
18.4k32453
answered 44 mins ago
parsiadparsiad
18.4k32453
answered 44 mins ago
parsiadparsiad
18.4k32453
answered 44 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
42 mins ago
1
$begingroup$
And also closed
$endgroup$
– Keen-ameteur
41 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
37 mins ago
$begingroup$
@BrevanEllefsen: +1, but I couldn't resist... :-)
$endgroup$
– parsiad
36 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
42 mins ago
1
$begingroup$
And also closed
$endgroup$
– Keen-ameteur
41 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
37 mins ago
$begingroup$
@BrevanEllefsen: +1, but I couldn't resist... :-)
$endgroup$
– parsiad
36 mins ago
$begingroup$
But R is an open set, the intersection is also R so it is still an open set
$endgroup$
– Tony Tong
42 mins ago
$begingroup$
But R is an open set, the intersection is also R so it is still an open set
$endgroup$
– Tony Tong
42 mins ago
1
1
$begingroup$
And also closed
$endgroup$
– Keen-ameteur
41 mins ago
$begingroup$
And also closed
$endgroup$
– Keen-ameteur
41 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
37 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
37 mins ago
$begingroup$
@BrevanEllefsen: +1, but I couldn't resist... :-)
$endgroup$
– parsiad
36 mins ago
$begingroup$
@BrevanEllefsen: +1, but I couldn't resist... :-)
$endgroup$
– parsiad
36 mins ago
add a comment |
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.
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
44 mins ago
$begingroup$
Oh it will get ${0}$
$endgroup$
– Tony Tong
39 mins ago