Examples for $ bar{A}cap bar{B}neqemptyset $, but $ bar{A}cap B=Acapbar{B}=emptyset $.












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Are there examples for sets $ A, Bsubset X $, where $ X $ is a topological spacce and $ A, B $ are its nonempty subsets, satisfying $ bar{A}cap bar{B}neqemptyset $, but $ bar{A}cap B=Acapbar{B}=emptyset $.




I came up with this question when I was reading Basic Topology(M.A.Armstrong). I am trying to gain more intuitions about the difference between conditions for connectedness and separated from one another in $ X $.




Note that a space $ X $ is connected if whenever it is decomposed as the union $ Acup B $ of two nonempty subsests then $ bar{A}cap Bneqemptyset $ or $ Acap bar{B}neqemptyset $.



And if $ A $ and $ B $ are subsets of a space $ X $, and if $ bar{A}capbar{B} $ is empty, we say that $ A $ and $ B $ are separated from one another in $ X $.







general-topology







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    Are there examples for sets $ A, Bsubset X $, where $ X $ is a topological spacce and $ A, B $ are its nonempty subsets, satisfying $ bar{A}cap bar{B}neqemptyset $, but $ bar{A}cap B=Acapbar{B}=emptyset $.




    I came up with this question when I was reading Basic Topology(M.A.Armstrong). I am trying to gain more intuitions about the difference between conditions for connectedness and separated from one another in $ X $.




    Note that a space $ X $ is connected if whenever it is decomposed as the union $ Acup B $ of two nonempty subsests then $ bar{A}cap Bneqemptyset $ or $ Acap bar{B}neqemptyset $.



    And if $ A $ and $ B $ are subsets of a space $ X $, and if $ bar{A}capbar{B} $ is empty, we say that $ A $ and $ B $ are separated from one another in $ X $.







    general-topology







    share|cite|improve this question









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      0












      0








      0





      $begingroup$



      Are there examples for sets $ A, Bsubset X $, where $ X $ is a topological spacce and $ A, B $ are its nonempty subsets, satisfying $ bar{A}cap bar{B}neqemptyset $, but $ bar{A}cap B=Acapbar{B}=emptyset $.




      I came up with this question when I was reading Basic Topology(M.A.Armstrong). I am trying to gain more intuitions about the difference between conditions for connectedness and separated from one another in $ X $.




      Note that a space $ X $ is connected if whenever it is decomposed as the union $ Acup B $ of two nonempty subsests then $ bar{A}cap Bneqemptyset $ or $ Acap bar{B}neqemptyset $.



      And if $ A $ and $ B $ are subsets of a space $ X $, and if $ bar{A}capbar{B} $ is empty, we say that $ A $ and $ B $ are separated from one another in $ X $.







      general-topology







      share|cite|improve this question









      $endgroup$





      Are there examples for sets $ A, Bsubset X $, where $ X $ is a topological spacce and $ A, B $ are its nonempty subsets, satisfying $ bar{A}cap bar{B}neqemptyset $, but $ bar{A}cap B=Acapbar{B}=emptyset $.




      I came up with this question when I was reading Basic Topology(M.A.Armstrong). I am trying to gain more intuitions about the difference between conditions for connectedness and separated from one another in $ X $.




      Note that a space $ X $ is connected if whenever it is decomposed as the union $ Acup B $ of two nonempty subsests then $ bar{A}cap Bneqemptyset $ or $ Acap bar{B}neqemptyset $.



      And if $ A $ and $ B $ are subsets of a space $ X $, and if $ bar{A}capbar{B} $ is empty, we say that $ A $ and $ B $ are separated from one another in $ X $.







      general-topology




      general-topology






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      share|cite|improve this question




      share|cite|improve this question










      asked Jan 2 at 2:45









      user549397user549397

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          1 Answer
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          $begingroup$

          In the ordinary real line, let $A=(0,1)$ and $B=(1,2)$, so that $1 in overline{A} cap overline{B}$.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            but $overline{A}bigcap B not= emptyset$
            $endgroup$
            – Joel Pereira
            Jan 2 at 5:21










          • $begingroup$
            @JoelPereira how so? $overline{A}=[0,1]$. What belongs to this as well as $(1,2)$, which consists of things strictly bigger than $1$?
            $endgroup$
            – Randall
            Jan 2 at 5:22












          • $begingroup$
            oh i read it as "A complement" as opposed to cl(A). my fault.
            $endgroup$
            – Joel Pereira
            Jan 2 at 5:24












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          active

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          3












          $begingroup$

          In the ordinary real line, let $A=(0,1)$ and $B=(1,2)$, so that $1 in overline{A} cap overline{B}$.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            but $overline{A}bigcap B not= emptyset$
            $endgroup$
            – Joel Pereira
            Jan 2 at 5:21










          • $begingroup$
            @JoelPereira how so? $overline{A}=[0,1]$. What belongs to this as well as $(1,2)$, which consists of things strictly bigger than $1$?
            $endgroup$
            – Randall
            Jan 2 at 5:22












          • $begingroup$
            oh i read it as "A complement" as opposed to cl(A). my fault.
            $endgroup$
            – Joel Pereira
            Jan 2 at 5:24
















          3












          $begingroup$

          In the ordinary real line, let $A=(0,1)$ and $B=(1,2)$, so that $1 in overline{A} cap overline{B}$.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            but $overline{A}bigcap B not= emptyset$
            $endgroup$
            – Joel Pereira
            Jan 2 at 5:21










          • $begingroup$
            @JoelPereira how so? $overline{A}=[0,1]$. What belongs to this as well as $(1,2)$, which consists of things strictly bigger than $1$?
            $endgroup$
            – Randall
            Jan 2 at 5:22












          • $begingroup$
            oh i read it as "A complement" as opposed to cl(A). my fault.
            $endgroup$
            – Joel Pereira
            Jan 2 at 5:24














          3












          3








          3





          $begingroup$

          In the ordinary real line, let $A=(0,1)$ and $B=(1,2)$, so that $1 in overline{A} cap overline{B}$.






          share|cite|improve this answer











          $endgroup$



          In the ordinary real line, let $A=(0,1)$ and $B=(1,2)$, so that $1 in overline{A} cap overline{B}$.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Jan 2 at 2:51

























          answered Jan 2 at 2:47









          RandallRandall

          10.6k11431




          10.6k11431












          • $begingroup$
            but $overline{A}bigcap B not= emptyset$
            $endgroup$
            – Joel Pereira
            Jan 2 at 5:21










          • $begingroup$
            @JoelPereira how so? $overline{A}=[0,1]$. What belongs to this as well as $(1,2)$, which consists of things strictly bigger than $1$?
            $endgroup$
            – Randall
            Jan 2 at 5:22












          • $begingroup$
            oh i read it as "A complement" as opposed to cl(A). my fault.
            $endgroup$
            – Joel Pereira
            Jan 2 at 5:24


















          • $begingroup$
            but $overline{A}bigcap B not= emptyset$
            $endgroup$
            – Joel Pereira
            Jan 2 at 5:21










          • $begingroup$
            @JoelPereira how so? $overline{A}=[0,1]$. What belongs to this as well as $(1,2)$, which consists of things strictly bigger than $1$?
            $endgroup$
            – Randall
            Jan 2 at 5:22












          • $begingroup$
            oh i read it as "A complement" as opposed to cl(A). my fault.
            $endgroup$
            – Joel Pereira
            Jan 2 at 5:24
















          $begingroup$
          but $overline{A}bigcap B not= emptyset$
          $endgroup$
          – Joel Pereira
          Jan 2 at 5:21




          $begingroup$
          but $overline{A}bigcap B not= emptyset$
          $endgroup$
          – Joel Pereira
          Jan 2 at 5:21












          $begingroup$
          @JoelPereira how so? $overline{A}=[0,1]$. What belongs to this as well as $(1,2)$, which consists of things strictly bigger than $1$?
          $endgroup$
          – Randall
          Jan 2 at 5:22






          $begingroup$
          @JoelPereira how so? $overline{A}=[0,1]$. What belongs to this as well as $(1,2)$, which consists of things strictly bigger than $1$?
          $endgroup$
          – Randall
          Jan 2 at 5:22














          $begingroup$
          oh i read it as "A complement" as opposed to cl(A). my fault.
          $endgroup$
          – Joel Pereira
          Jan 2 at 5:24




          $begingroup$
          oh i read it as "A complement" as opposed to cl(A). my fault.
          $endgroup$
          – Joel Pereira
          Jan 2 at 5:24


















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