Give an example of subset $B$ of the real line $mathbb{R}$ so the subsets $A$, $Int(A)$, $overline{A}$, dont...












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What is an example of a subset $A$ of the real line $mathbb{R}$ (equipped with the standard metric topology), such that
the subsets $A$, $Int(A)$, $overline{A}$, $overline{Int(A)}$ and Int($overline{A}$) are pairwise different?






general-topology







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closed as off-topic by amWhy, Brahadeesh, DRF, Lord_Farin, KReiser Dec 4 '18 at 1:24


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 improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – amWhy, Brahadeesh, DRF, Lord_Farin, KReiser

If this question can be reworded to fit the rules in the help center, please edit the question.









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    Welcome to mathSE. What have you tried? Where did you get stuck? Please add some information about your work done. As is the question will be closed.
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0












$begingroup$


What is an example of a subset $A$ of the real line $mathbb{R}$ (equipped with the standard metric topology), such that
the subsets $A$, $Int(A)$, $overline{A}$, $overline{Int(A)}$ and Int($overline{A}$) are pairwise different?






general-topology







share|cite|improve this question











$endgroup$



closed as off-topic by amWhy, Brahadeesh, DRF, Lord_Farin, KReiser Dec 4 '18 at 1:24


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 improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – amWhy, Brahadeesh, DRF, Lord_Farin, KReiser

If this question can be reworded to fit the rules in the help center, please edit the question.









  • 1




    $begingroup$
    Welcome to mathSE. What have you tried? Where did you get stuck? Please add some information about your work done. As is the question will be closed.
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    – DRF
    Dec 3 '18 at 14:12














0












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0





$begingroup$


What is an example of a subset $A$ of the real line $mathbb{R}$ (equipped with the standard metric topology), such that
the subsets $A$, $Int(A)$, $overline{A}$, $overline{Int(A)}$ and Int($overline{A}$) are pairwise different?






general-topology







share|cite|improve this question











$endgroup$




What is an example of a subset $A$ of the real line $mathbb{R}$ (equipped with the standard metric topology), such that
the subsets $A$, $Int(A)$, $overline{A}$, $overline{Int(A)}$ and Int($overline{A}$) are pairwise different?






general-topology




general-topology






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share|cite|improve this question













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share|cite|improve this question








edited Dec 6 '18 at 8:54







Esteban Cambiasso

















asked Dec 3 '18 at 11:35









Esteban CambiassoEsteban Cambiasso

103




103




closed as off-topic by amWhy, Brahadeesh, DRF, Lord_Farin, KReiser Dec 4 '18 at 1:24


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 improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – amWhy, Brahadeesh, DRF, Lord_Farin, KReiser

If this question can be reworded to fit the rules in the help center, please edit the question.




closed as off-topic by amWhy, Brahadeesh, DRF, Lord_Farin, KReiser Dec 4 '18 at 1:24


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 improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – amWhy, Brahadeesh, DRF, Lord_Farin, KReiser

If this question can be reworded to fit the rules in the help center, please edit the question.








  • 1




    $begingroup$
    Welcome to mathSE. What have you tried? Where did you get stuck? Please add some information about your work done. As is the question will be closed.
    $endgroup$
    – DRF
    Dec 3 '18 at 14:12














  • 1




    $begingroup$
    Welcome to mathSE. What have you tried? Where did you get stuck? Please add some information about your work done. As is the question will be closed.
    $endgroup$
    – DRF
    Dec 3 '18 at 14:12








1




1




$begingroup$
Welcome to mathSE. What have you tried? Where did you get stuck? Please add some information about your work done. As is the question will be closed.
$endgroup$
– DRF
Dec 3 '18 at 14:12




$begingroup$
Welcome to mathSE. What have you tried? Where did you get stuck? Please add some information about your work done. As is the question will be closed.
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– DRF
Dec 3 '18 at 14:12










2 Answers
2






active

oldest

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Let $B=Bbb Qcap(1,2),$ and let $A=(0,1)cup B.$






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  • $begingroup$
    ok, we got the same idea ahah (Typing latex with a phone is harder though)
    $endgroup$
    – Antonio Alfieri
    Dec 3 '18 at 12:01












  • $begingroup$
    True! Hence my brevity. ;-)
    $endgroup$
    – Cameron Buie
    Dec 3 '18 at 12:21



















2












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Take $A=((0,1) cap mathbb{Q}) cup (2,3)$. The interior of A is $(2,3)$, the closure of A is $[0,1] cup [2,3]$, the closure of the interior of A is $[2,3]$, and the interior of the closure of A is $(0,1) cup (2,3)$.






share|cite|improve this answer









$endgroup$




















    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    3












    $begingroup$

    Let $B=Bbb Qcap(1,2),$ and let $A=(0,1)cup B.$






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      ok, we got the same idea ahah (Typing latex with a phone is harder though)
      $endgroup$
      – Antonio Alfieri
      Dec 3 '18 at 12:01












    • $begingroup$
      True! Hence my brevity. ;-)
      $endgroup$
      – Cameron Buie
      Dec 3 '18 at 12:21
















    3












    $begingroup$

    Let $B=Bbb Qcap(1,2),$ and let $A=(0,1)cup B.$






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      ok, we got the same idea ahah (Typing latex with a phone is harder though)
      $endgroup$
      – Antonio Alfieri
      Dec 3 '18 at 12:01












    • $begingroup$
      True! Hence my brevity. ;-)
      $endgroup$
      – Cameron Buie
      Dec 3 '18 at 12:21














    3












    3








    3





    $begingroup$

    Let $B=Bbb Qcap(1,2),$ and let $A=(0,1)cup B.$






    share|cite|improve this answer









    $endgroup$



    Let $B=Bbb Qcap(1,2),$ and let $A=(0,1)cup B.$







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered Dec 3 '18 at 11:39









    Cameron BuieCameron Buie

    85.1k771155




    85.1k771155












    • $begingroup$
      ok, we got the same idea ahah (Typing latex with a phone is harder though)
      $endgroup$
      – Antonio Alfieri
      Dec 3 '18 at 12:01












    • $begingroup$
      True! Hence my brevity. ;-)
      $endgroup$
      – Cameron Buie
      Dec 3 '18 at 12:21


















    • $begingroup$
      ok, we got the same idea ahah (Typing latex with a phone is harder though)
      $endgroup$
      – Antonio Alfieri
      Dec 3 '18 at 12:01












    • $begingroup$
      True! Hence my brevity. ;-)
      $endgroup$
      – Cameron Buie
      Dec 3 '18 at 12:21
















    $begingroup$
    ok, we got the same idea ahah (Typing latex with a phone is harder though)
    $endgroup$
    – Antonio Alfieri
    Dec 3 '18 at 12:01






    $begingroup$
    ok, we got the same idea ahah (Typing latex with a phone is harder though)
    $endgroup$
    – Antonio Alfieri
    Dec 3 '18 at 12:01














    $begingroup$
    True! Hence my brevity. ;-)
    $endgroup$
    – Cameron Buie
    Dec 3 '18 at 12:21




    $begingroup$
    True! Hence my brevity. ;-)
    $endgroup$
    – Cameron Buie
    Dec 3 '18 at 12:21











    2












    $begingroup$

    Take $A=((0,1) cap mathbb{Q}) cup (2,3)$. The interior of A is $(2,3)$, the closure of A is $[0,1] cup [2,3]$, the closure of the interior of A is $[2,3]$, and the interior of the closure of A is $(0,1) cup (2,3)$.






    share|cite|improve this answer









    $endgroup$


















      2












      $begingroup$

      Take $A=((0,1) cap mathbb{Q}) cup (2,3)$. The interior of A is $(2,3)$, the closure of A is $[0,1] cup [2,3]$, the closure of the interior of A is $[2,3]$, and the interior of the closure of A is $(0,1) cup (2,3)$.






      share|cite|improve this answer









      $endgroup$
















        2












        2








        2





        $begingroup$

        Take $A=((0,1) cap mathbb{Q}) cup (2,3)$. The interior of A is $(2,3)$, the closure of A is $[0,1] cup [2,3]$, the closure of the interior of A is $[2,3]$, and the interior of the closure of A is $(0,1) cup (2,3)$.






        share|cite|improve this answer









        $endgroup$



        Take $A=((0,1) cap mathbb{Q}) cup (2,3)$. The interior of A is $(2,3)$, the closure of A is $[0,1] cup [2,3]$, the closure of the interior of A is $[2,3]$, and the interior of the closure of A is $(0,1) cup (2,3)$.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 3 '18 at 11:59









        Antonio AlfieriAntonio Alfieri

        1,197412




        1,197412















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