Question on point set topology












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Does there exist a closed set which is an intersection of a collection of infinite open sets?






analysis







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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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  • 5




    $begingroup$
    Intersect $(-tfrac{1}{n}, tfrac{1}{n})$ for $n = 1, 2, ldots$ and consider what set you get
    $endgroup$
    – Brevan Ellefsen
    36 mins ago










  • $begingroup$
    Oh it will get ${0}$
    $endgroup$
    – Tony Tong
    31 mins ago
















0












$begingroup$


Does there exist a closed set which is an intersection of a collection of infinite open sets?






analysis







share|cite|improve this question







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
    36 mins ago










  • $begingroup$
    Oh it will get ${0}$
    $endgroup$
    – Tony Tong
    31 mins ago














0












0








0





$begingroup$


Does there exist a closed set which is an intersection of a collection of infinite open sets?






analysis







share|cite|improve this question







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






share|cite|improve this question







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.











share|cite|improve this question







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.









share|cite|improve this question




share|cite|improve this question






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 39 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
    36 mins ago










  • $begingroup$
    Oh it will get ${0}$
    $endgroup$
    – Tony Tong
    31 mins ago














  • 5




    $begingroup$
    Intersect $(-tfrac{1}{n}, tfrac{1}{n})$ for $n = 1, 2, ldots$ and consider what set you get
    $endgroup$
    – Brevan Ellefsen
    36 mins ago










  • $begingroup$
    Oh it will get ${0}$
    $endgroup$
    – Tony Tong
    31 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
36 mins ago




$begingroup$
Intersect $(-tfrac{1}{n}, tfrac{1}{n})$ for $n = 1, 2, ldots$ and consider what set you get
$endgroup$
– Brevan Ellefsen
36 mins ago












$begingroup$
Oh it will get ${0}$
$endgroup$
– Tony Tong
31 mins ago




$begingroup$
Oh it will get ${0}$
$endgroup$
– Tony Tong
31 mins ago










1 Answer
1






active

oldest

votes


















4












$begingroup$

$$mathbb{R}capmathbb{R}capmathbb{R}capcdots$$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    But R is an open set, the intersection is also R so it is still an open set
    $endgroup$
    – Tony Tong
    35 mins ago








  • 1




    $begingroup$
    And also closed
    $endgroup$
    – Keen-ameteur
    33 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
    29 mins ago












  • $begingroup$
    @BrevanEllefsen: +1, but I couldn't resist... :-)
    $endgroup$
    – parsiad
    29 mins ago













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1 Answer
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1 Answer
1






active

oldest

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active

oldest

votes






active

oldest

votes









4












$begingroup$

$$mathbb{R}capmathbb{R}capmathbb{R}capcdots$$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    But R is an open set, the intersection is also R so it is still an open set
    $endgroup$
    – Tony Tong
    35 mins ago








  • 1




    $begingroup$
    And also closed
    $endgroup$
    – Keen-ameteur
    33 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
    29 mins ago












  • $begingroup$
    @BrevanEllefsen: +1, but I couldn't resist... :-)
    $endgroup$
    – parsiad
    29 mins ago


















4












$begingroup$

$$mathbb{R}capmathbb{R}capmathbb{R}capcdots$$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    But R is an open set, the intersection is also R so it is still an open set
    $endgroup$
    – Tony Tong
    35 mins ago








  • 1




    $begingroup$
    And also closed
    $endgroup$
    – Keen-ameteur
    33 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
    29 mins ago












  • $begingroup$
    @BrevanEllefsen: +1, but I couldn't resist... :-)
    $endgroup$
    – parsiad
    29 mins ago
















4












4








4





$begingroup$

$$mathbb{R}capmathbb{R}capmathbb{R}capcdots$$






share|cite|improve this answer









$endgroup$



$$mathbb{R}capmathbb{R}capmathbb{R}capcdots$$







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered 36 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
    35 mins ago








  • 1




    $begingroup$
    And also closed
    $endgroup$
    – Keen-ameteur
    33 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
    29 mins ago












  • $begingroup$
    @BrevanEllefsen: +1, but I couldn't resist... :-)
    $endgroup$
    – parsiad
    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
    35 mins ago








  • 1




    $begingroup$
    And also closed
    $endgroup$
    – Keen-ameteur
    33 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
    29 mins ago












  • $begingroup$
    @BrevanEllefsen: +1, but I couldn't resist... :-)
    $endgroup$
    – parsiad
    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
35 mins ago






$begingroup$
But R is an open set, the intersection is also R so it is still an open set
$endgroup$
– Tony Tong
35 mins ago






1




1




$begingroup$
And also closed
$endgroup$
– Keen-ameteur
33 mins ago




$begingroup$
And also closed
$endgroup$
– Keen-ameteur
33 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
29 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
29 mins ago














$begingroup$
@BrevanEllefsen: +1, but I couldn't resist... :-)
$endgroup$
– parsiad
29 mins ago






$begingroup$
@BrevanEllefsen: +1, but I couldn't resist... :-)
$endgroup$
– parsiad
29 mins ago












Tony Tong is a new contributor. Be nice, and check out our Code of Conduct.










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