Is this set meager? $A = {xin mathbb{R}: exists c>0, |x-j2^{-k}|geq c2^{-k}, forall jin mathbb{Z}, kgeq 0...












2












$begingroup$


We define the subset $Asubset mathbb{R}$ as follows: $xin A Longleftrightarrow$ if $exists c>0$ so that
$$ |x-j2^{-k}|geq c2^{-k} $$
holds $forall jin mathbb{Z}$ and integers $kgeq 0$. Show that $A$ is meager and dense.




I am completely lost on this one. It looks like it is saying that $xin A$ if the difference between $x$ and any dyadic rational can be made greater than $c2^{-k}$. But I have no intuition for this at all.



Anyone have a hint (not a solution) for how to approach this problem?




Edit:



So I tried to see if for each $k$, say $0,1,2,...$, I could create a set $A_k$ which is nowhere dense. But $A_0 = mathbb{R}/mathbb{Z}$ is not nowhere dense. So that approach isn't working unless I am confused.






real-analysis general-topology functional-analysis complete-spaces







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$endgroup$

















    2












    $begingroup$


    We define the subset $Asubset mathbb{R}$ as follows: $xin A Longleftrightarrow$ if $exists c>0$ so that
    $$ |x-j2^{-k}|geq c2^{-k} $$
    holds $forall jin mathbb{Z}$ and integers $kgeq 0$. Show that $A$ is meager and dense.




    I am completely lost on this one. It looks like it is saying that $xin A$ if the difference between $x$ and any dyadic rational can be made greater than $c2^{-k}$. But I have no intuition for this at all.



    Anyone have a hint (not a solution) for how to approach this problem?




    Edit:



    So I tried to see if for each $k$, say $0,1,2,...$, I could create a set $A_k$ which is nowhere dense. But $A_0 = mathbb{R}/mathbb{Z}$ is not nowhere dense. So that approach isn't working unless I am confused.






    real-analysis general-topology functional-analysis complete-spaces







    share|cite|improve this question











    $endgroup$















      2












      2








      2


      2



      $begingroup$


      We define the subset $Asubset mathbb{R}$ as follows: $xin A Longleftrightarrow$ if $exists c>0$ so that
      $$ |x-j2^{-k}|geq c2^{-k} $$
      holds $forall jin mathbb{Z}$ and integers $kgeq 0$. Show that $A$ is meager and dense.




      I am completely lost on this one. It looks like it is saying that $xin A$ if the difference between $x$ and any dyadic rational can be made greater than $c2^{-k}$. But I have no intuition for this at all.



      Anyone have a hint (not a solution) for how to approach this problem?




      Edit:



      So I tried to see if for each $k$, say $0,1,2,...$, I could create a set $A_k$ which is nowhere dense. But $A_0 = mathbb{R}/mathbb{Z}$ is not nowhere dense. So that approach isn't working unless I am confused.






      real-analysis general-topology functional-analysis complete-spaces







      share|cite|improve this question











      $endgroup$




      We define the subset $Asubset mathbb{R}$ as follows: $xin A Longleftrightarrow$ if $exists c>0$ so that
      $$ |x-j2^{-k}|geq c2^{-k} $$
      holds $forall jin mathbb{Z}$ and integers $kgeq 0$. Show that $A$ is meager and dense.




      I am completely lost on this one. It looks like it is saying that $xin A$ if the difference between $x$ and any dyadic rational can be made greater than $c2^{-k}$. But I have no intuition for this at all.



      Anyone have a hint (not a solution) for how to approach this problem?




      Edit:



      So I tried to see if for each $k$, say $0,1,2,...$, I could create a set $A_k$ which is nowhere dense. But $A_0 = mathbb{R}/mathbb{Z}$ is not nowhere dense. So that approach isn't working unless I am confused.






      real-analysis general-topology functional-analysis complete-spaces




      real-analysis general-topology functional-analysis complete-spaces






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




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      edited Dec 1 '18 at 9:11







      Joe Man Analysis

















      asked Dec 1 '18 at 5:03









      Joe Man AnalysisJoe Man Analysis

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      33419






















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          $begingroup$

          We have $A=bigcup A_n$, where $$A_n={xin mathbb{R}: |x-j2^{-k}|geq 2^{-n-k}, forall jin mathbb{Z}, kgeq 0 }.$$



          Spoiler:




          So it suffices to show that each $A_n$ is meager. Since $$A_n=bigcap_{jinBbb Z, >! kge 0} Bbb Rsetminus (j2^{-k}-2^{-n-k}, j2^{-k}+2^{-n-k}),$$
          it is closed. Since $A_n$ is disjoint with the set of diadic rationals, the set $A_n$ is nowhere dense.







          share|cite|improve this answer











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            $begingroup$

            We have $A=bigcup A_n$, where $$A_n={xin mathbb{R}: |x-j2^{-k}|geq 2^{-n-k}, forall jin mathbb{Z}, kgeq 0 }.$$



            Spoiler:




            So it suffices to show that each $A_n$ is meager. Since $$A_n=bigcap_{jinBbb Z, >! kge 0} Bbb Rsetminus (j2^{-k}-2^{-n-k}, j2^{-k}+2^{-n-k}),$$
            it is closed. Since $A_n$ is disjoint with the set of diadic rationals, the set $A_n$ is nowhere dense.







            share|cite|improve this answer











            $endgroup$


















              3












              $begingroup$

              We have $A=bigcup A_n$, where $$A_n={xin mathbb{R}: |x-j2^{-k}|geq 2^{-n-k}, forall jin mathbb{Z}, kgeq 0 }.$$



              Spoiler:




              So it suffices to show that each $A_n$ is meager. Since $$A_n=bigcap_{jinBbb Z, >! kge 0} Bbb Rsetminus (j2^{-k}-2^{-n-k}, j2^{-k}+2^{-n-k}),$$
              it is closed. Since $A_n$ is disjoint with the set of diadic rationals, the set $A_n$ is nowhere dense.







              share|cite|improve this answer











              $endgroup$
















                3












                3








                3





                $begingroup$

                We have $A=bigcup A_n$, where $$A_n={xin mathbb{R}: |x-j2^{-k}|geq 2^{-n-k}, forall jin mathbb{Z}, kgeq 0 }.$$



                Spoiler:




                So it suffices to show that each $A_n$ is meager. Since $$A_n=bigcap_{jinBbb Z, >! kge 0} Bbb Rsetminus (j2^{-k}-2^{-n-k}, j2^{-k}+2^{-n-k}),$$
                it is closed. Since $A_n$ is disjoint with the set of diadic rationals, the set $A_n$ is nowhere dense.







                share|cite|improve this answer











                $endgroup$



                We have $A=bigcup A_n$, where $$A_n={xin mathbb{R}: |x-j2^{-k}|geq 2^{-n-k}, forall jin mathbb{Z}, kgeq 0 }.$$



                Spoiler:




                So it suffices to show that each $A_n$ is meager. Since $$A_n=bigcap_{jinBbb Z, >! kge 0} Bbb Rsetminus (j2^{-k}-2^{-n-k}, j2^{-k}+2^{-n-k}),$$
                it is closed. Since $A_n$ is disjoint with the set of diadic rationals, the set $A_n$ is nowhere dense.








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                share|cite|improve this answer








                edited Dec 2 '18 at 6:56

























                answered Dec 2 '18 at 6:50









                Alex RavskyAlex Ravsky

                39.9k32282




                39.9k32282






























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