some confusion about open subset?











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Consider the set of rational number $mathbb{Q}$ as a subset of $mathbb{R}$ with the usual metric. Let $K = [ sqrt 2, sqrt 3] cap mathbb{Q}$.



I have some confusion in my mind that is



Is $K$ is an open subset of $mathbb{Q}$ ?



My attempt : my answer is No,



$K=[sqrt 2, sqrt 3]cap Bbb{Q}={q in Bbb{Q}|sqrt 2< q< sqrt 3}$ where$[sqrt 2, sqrt 3]$ is closed in $Bbb{R}$.



From this I can conclude that K is not open subset of $mathbb{Q}$



Is it True ?










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    up vote
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    down vote

    favorite












    Consider the set of rational number $mathbb{Q}$ as a subset of $mathbb{R}$ with the usual metric. Let $K = [ sqrt 2, sqrt 3] cap mathbb{Q}$.



    I have some confusion in my mind that is



    Is $K$ is an open subset of $mathbb{Q}$ ?



    My attempt : my answer is No,



    $K=[sqrt 2, sqrt 3]cap Bbb{Q}={q in Bbb{Q}|sqrt 2< q< sqrt 3}$ where$[sqrt 2, sqrt 3]$ is closed in $Bbb{R}$.



    From this I can conclude that K is not open subset of $mathbb{Q}$



    Is it True ?










    share|cite|improve this question


























      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      Consider the set of rational number $mathbb{Q}$ as a subset of $mathbb{R}$ with the usual metric. Let $K = [ sqrt 2, sqrt 3] cap mathbb{Q}$.



      I have some confusion in my mind that is



      Is $K$ is an open subset of $mathbb{Q}$ ?



      My attempt : my answer is No,



      $K=[sqrt 2, sqrt 3]cap Bbb{Q}={q in Bbb{Q}|sqrt 2< q< sqrt 3}$ where$[sqrt 2, sqrt 3]$ is closed in $Bbb{R}$.



      From this I can conclude that K is not open subset of $mathbb{Q}$



      Is it True ?










      share|cite|improve this question















      Consider the set of rational number $mathbb{Q}$ as a subset of $mathbb{R}$ with the usual metric. Let $K = [ sqrt 2, sqrt 3] cap mathbb{Q}$.



      I have some confusion in my mind that is



      Is $K$ is an open subset of $mathbb{Q}$ ?



      My attempt : my answer is No,



      $K=[sqrt 2, sqrt 3]cap Bbb{Q}={q in Bbb{Q}|sqrt 2< q< sqrt 3}$ where$[sqrt 2, sqrt 3]$ is closed in $Bbb{R}$.



      From this I can conclude that K is not open subset of $mathbb{Q}$



      Is it True ?







      general-topology proof-verification compactness






      share|cite|improve this question















      share|cite|improve this question













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      edited 9 hours ago









      José Carlos Santos

      146k22116216




      146k22116216










      asked 9 hours ago









      jasmine

      1,446416




      1,446416






















          5 Answers
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          up vote
          5
          down vote



          accepted










          Yes, $K$ is an open subset of $mathbb Q$, since $K=left(sqrt2,sqrt3right)capmathbb Q$ and $left(sqrt2,sqrt3right)$ is an open subset of $mathbb R$.






          share|cite|improve this answer




























            up vote
            13
            down vote













            No that's wrong. The fact that a set is closed doesn't mean it is not open!



            In fact $K$ is also open because it equals to $(sqrt{2},sqrt{3})cap mathbb{Q}$.



            Side note: The space $mathbb{Q}$ with the topology induced by $mathbb{R}$ is "totally disconnected" this means that it has "many" sets which are both closed and open.






            share|cite|improve this answer





















            • A set is not a door.
              – Arno
              1 hour ago


















            up vote
            3
            down vote













            A set can be both open and closed at the same time (such sets are called clopen), and just because you've shown that $[sqrt2, sqrt3]cap Bbb Q$ is closed in $Bbb Q$, that doen't mean it isn't open.



            Look at the definition of open in the subspace topology, and se whether $[sqrt2, sqrt3]cap Bbb Q$ is such a set or not.






            share|cite|improve this answer




























              up vote
              0
              down vote













              With the metric $d(x,y)=|x-y|$ on $Bbb Q$ and the topology on $Bbb Q$ generated by $d$: For $qin K$ let $r(q)=min (q-sqrt 2,,sqrt 3 -q,).$ Let $K(q)={q'in Bbb Q: d(q',q)<r(q)}.$



              Then $K(q)$ is open in $Bbb Q$ and $qin K(q)subset K.$



              So $cup_{qin K}K(q)$ is open in $Bbb Q.$ And we have $K=cup_{qin K}
              ,{q}subset cup_{qin K},K(q)subset cup_{qin K},K=K,$
              so $K=cup_{qin K}K(q)$ is open in $Bbb Q.$



              An easily overlooked point about this Q:



              (i). Let $T$ be a topology on a set $X$ and let $Y subset X.$ The subspace topology $T|Y$ on $Y$ is defined as $T|Y={tcap Y:tin T}. $ If $B$ is a base (basis) for $T$ then $B|Y={bcap Y: bin B}$ is a base for $T|Y.$



              (ii).Suppose $T$ is generated by a metric $d$ on $X,$ so that the set $B$ of open $d$-balls of $X$ is a base for $T$. So $B|Y$ is a base for $T|Y.$



              BUT in general $B|Y$ may NOT be the set of open $d$-balls of $Y.$ An open $d$-ball of $Y$ is $B_d^Y(y,r)={y'in Y: d(y',y)<r},$ for some $yin Y, $ which does belong to $B|Y,$ but there may be other members of $B|Y.$



              For example in your Q, with $X=Bbb R$ and $Y=Bbb Q$ and $d(u,v)=|u-v|,$ the set $K$ belongs to $B|Y$ but is not an open ball of $Bbb Q$ because $(sqrt 2 +sqrt 3)/2not in Bbb Q.$



              (iii). In the general case, for metric spacess it is a useful, widely used result that every member of $B|Y$ is a union of open $d$-balls of $Y,$ so the subspace topology $T|Y,$ as a subspace of $X,$ co-incides with the topology on $Y$ generated by the metric $d|_{Ytimes Y}.$






              share|cite|improve this answer




























                up vote
                0
                down vote














                $K=[sqrt 2,sqrt 3]∩mathbb{Q}={qin mathbb{Q}|sqrt 2<q<sqrt 3}$ where$[sqrt 2,sqrt 3]$ is closed in R.



                From this I can conclude that K is not open subset of Q




                You're really not making it clear what your reasoning is. You seem to mostly just be restating the problem statement. Reading between the lines, your argument seems to be:




                1. K is an intersection between a closed set and a closed set.


                2. K is therefore closed.


                3. Therefore K is not open.



                The third statement is wrong; that a set is closed doesn't mean it's open. Presenting your argument explicitly helps others, and hopefully yourself, see what's wrong with it.



                If we have the open ball topology, then since $sqrt2 <q$, we know that there is "space" between $sqrt 2$ and $q$, and similarly for $sqrt3$. So given any $q$, we can take $epsilon_1$ to be half the distance between $sqrt2$ and $q$, $epsilon_2$ to be half the distance between $sqrt3$ and $q$, and $epsilon$ to be the minimum of $epsilon_1$ and $epsilon_2$. Then everything withing $epsilon$ of $q$ is in K, so $q$ is interior, and since $q$ is arbitrary, K is open.






                share|cite|improve this answer























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                  5 Answers
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                  up vote
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                  down vote



                  accepted










                  Yes, $K$ is an open subset of $mathbb Q$, since $K=left(sqrt2,sqrt3right)capmathbb Q$ and $left(sqrt2,sqrt3right)$ is an open subset of $mathbb R$.






                  share|cite|improve this answer

























                    up vote
                    5
                    down vote



                    accepted










                    Yes, $K$ is an open subset of $mathbb Q$, since $K=left(sqrt2,sqrt3right)capmathbb Q$ and $left(sqrt2,sqrt3right)$ is an open subset of $mathbb R$.






                    share|cite|improve this answer























                      up vote
                      5
                      down vote



                      accepted







                      up vote
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                      down vote



                      accepted






                      Yes, $K$ is an open subset of $mathbb Q$, since $K=left(sqrt2,sqrt3right)capmathbb Q$ and $left(sqrt2,sqrt3right)$ is an open subset of $mathbb R$.






                      share|cite|improve this answer












                      Yes, $K$ is an open subset of $mathbb Q$, since $K=left(sqrt2,sqrt3right)capmathbb Q$ and $left(sqrt2,sqrt3right)$ is an open subset of $mathbb R$.







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                      answered 9 hours ago









                      José Carlos Santos

                      146k22116216




                      146k22116216






















                          up vote
                          13
                          down vote













                          No that's wrong. The fact that a set is closed doesn't mean it is not open!



                          In fact $K$ is also open because it equals to $(sqrt{2},sqrt{3})cap mathbb{Q}$.



                          Side note: The space $mathbb{Q}$ with the topology induced by $mathbb{R}$ is "totally disconnected" this means that it has "many" sets which are both closed and open.






                          share|cite|improve this answer





















                          • A set is not a door.
                            – Arno
                            1 hour ago















                          up vote
                          13
                          down vote













                          No that's wrong. The fact that a set is closed doesn't mean it is not open!



                          In fact $K$ is also open because it equals to $(sqrt{2},sqrt{3})cap mathbb{Q}$.



                          Side note: The space $mathbb{Q}$ with the topology induced by $mathbb{R}$ is "totally disconnected" this means that it has "many" sets which are both closed and open.






                          share|cite|improve this answer





















                          • A set is not a door.
                            – Arno
                            1 hour ago













                          up vote
                          13
                          down vote










                          up vote
                          13
                          down vote









                          No that's wrong. The fact that a set is closed doesn't mean it is not open!



                          In fact $K$ is also open because it equals to $(sqrt{2},sqrt{3})cap mathbb{Q}$.



                          Side note: The space $mathbb{Q}$ with the topology induced by $mathbb{R}$ is "totally disconnected" this means that it has "many" sets which are both closed and open.






                          share|cite|improve this answer












                          No that's wrong. The fact that a set is closed doesn't mean it is not open!



                          In fact $K$ is also open because it equals to $(sqrt{2},sqrt{3})cap mathbb{Q}$.



                          Side note: The space $mathbb{Q}$ with the topology induced by $mathbb{R}$ is "totally disconnected" this means that it has "many" sets which are both closed and open.







                          share|cite|improve this answer












                          share|cite|improve this answer



                          share|cite|improve this answer










                          answered 9 hours ago









                          Yanko

                          5,520723




                          5,520723












                          • A set is not a door.
                            – Arno
                            1 hour ago


















                          • A set is not a door.
                            – Arno
                            1 hour ago
















                          A set is not a door.
                          – Arno
                          1 hour ago




                          A set is not a door.
                          – Arno
                          1 hour ago










                          up vote
                          3
                          down vote













                          A set can be both open and closed at the same time (such sets are called clopen), and just because you've shown that $[sqrt2, sqrt3]cap Bbb Q$ is closed in $Bbb Q$, that doen't mean it isn't open.



                          Look at the definition of open in the subspace topology, and se whether $[sqrt2, sqrt3]cap Bbb Q$ is such a set or not.






                          share|cite|improve this answer

























                            up vote
                            3
                            down vote













                            A set can be both open and closed at the same time (such sets are called clopen), and just because you've shown that $[sqrt2, sqrt3]cap Bbb Q$ is closed in $Bbb Q$, that doen't mean it isn't open.



                            Look at the definition of open in the subspace topology, and se whether $[sqrt2, sqrt3]cap Bbb Q$ is such a set or not.






                            share|cite|improve this answer























                              up vote
                              3
                              down vote










                              up vote
                              3
                              down vote









                              A set can be both open and closed at the same time (such sets are called clopen), and just because you've shown that $[sqrt2, sqrt3]cap Bbb Q$ is closed in $Bbb Q$, that doen't mean it isn't open.



                              Look at the definition of open in the subspace topology, and se whether $[sqrt2, sqrt3]cap Bbb Q$ is such a set or not.






                              share|cite|improve this answer












                              A set can be both open and closed at the same time (such sets are called clopen), and just because you've shown that $[sqrt2, sqrt3]cap Bbb Q$ is closed in $Bbb Q$, that doen't mean it isn't open.



                              Look at the definition of open in the subspace topology, and se whether $[sqrt2, sqrt3]cap Bbb Q$ is such a set or not.







                              share|cite|improve this answer












                              share|cite|improve this answer



                              share|cite|improve this answer










                              answered 9 hours ago









                              Arthur

                              110k7104186




                              110k7104186






















                                  up vote
                                  0
                                  down vote













                                  With the metric $d(x,y)=|x-y|$ on $Bbb Q$ and the topology on $Bbb Q$ generated by $d$: For $qin K$ let $r(q)=min (q-sqrt 2,,sqrt 3 -q,).$ Let $K(q)={q'in Bbb Q: d(q',q)<r(q)}.$



                                  Then $K(q)$ is open in $Bbb Q$ and $qin K(q)subset K.$



                                  So $cup_{qin K}K(q)$ is open in $Bbb Q.$ And we have $K=cup_{qin K}
                                  ,{q}subset cup_{qin K},K(q)subset cup_{qin K},K=K,$
                                  so $K=cup_{qin K}K(q)$ is open in $Bbb Q.$



                                  An easily overlooked point about this Q:



                                  (i). Let $T$ be a topology on a set $X$ and let $Y subset X.$ The subspace topology $T|Y$ on $Y$ is defined as $T|Y={tcap Y:tin T}. $ If $B$ is a base (basis) for $T$ then $B|Y={bcap Y: bin B}$ is a base for $T|Y.$



                                  (ii).Suppose $T$ is generated by a metric $d$ on $X,$ so that the set $B$ of open $d$-balls of $X$ is a base for $T$. So $B|Y$ is a base for $T|Y.$



                                  BUT in general $B|Y$ may NOT be the set of open $d$-balls of $Y.$ An open $d$-ball of $Y$ is $B_d^Y(y,r)={y'in Y: d(y',y)<r},$ for some $yin Y, $ which does belong to $B|Y,$ but there may be other members of $B|Y.$



                                  For example in your Q, with $X=Bbb R$ and $Y=Bbb Q$ and $d(u,v)=|u-v|,$ the set $K$ belongs to $B|Y$ but is not an open ball of $Bbb Q$ because $(sqrt 2 +sqrt 3)/2not in Bbb Q.$



                                  (iii). In the general case, for metric spacess it is a useful, widely used result that every member of $B|Y$ is a union of open $d$-balls of $Y,$ so the subspace topology $T|Y,$ as a subspace of $X,$ co-incides with the topology on $Y$ generated by the metric $d|_{Ytimes Y}.$






                                  share|cite|improve this answer

























                                    up vote
                                    0
                                    down vote













                                    With the metric $d(x,y)=|x-y|$ on $Bbb Q$ and the topology on $Bbb Q$ generated by $d$: For $qin K$ let $r(q)=min (q-sqrt 2,,sqrt 3 -q,).$ Let $K(q)={q'in Bbb Q: d(q',q)<r(q)}.$



                                    Then $K(q)$ is open in $Bbb Q$ and $qin K(q)subset K.$



                                    So $cup_{qin K}K(q)$ is open in $Bbb Q.$ And we have $K=cup_{qin K}
                                    ,{q}subset cup_{qin K},K(q)subset cup_{qin K},K=K,$
                                    so $K=cup_{qin K}K(q)$ is open in $Bbb Q.$



                                    An easily overlooked point about this Q:



                                    (i). Let $T$ be a topology on a set $X$ and let $Y subset X.$ The subspace topology $T|Y$ on $Y$ is defined as $T|Y={tcap Y:tin T}. $ If $B$ is a base (basis) for $T$ then $B|Y={bcap Y: bin B}$ is a base for $T|Y.$



                                    (ii).Suppose $T$ is generated by a metric $d$ on $X,$ so that the set $B$ of open $d$-balls of $X$ is a base for $T$. So $B|Y$ is a base for $T|Y.$



                                    BUT in general $B|Y$ may NOT be the set of open $d$-balls of $Y.$ An open $d$-ball of $Y$ is $B_d^Y(y,r)={y'in Y: d(y',y)<r},$ for some $yin Y, $ which does belong to $B|Y,$ but there may be other members of $B|Y.$



                                    For example in your Q, with $X=Bbb R$ and $Y=Bbb Q$ and $d(u,v)=|u-v|,$ the set $K$ belongs to $B|Y$ but is not an open ball of $Bbb Q$ because $(sqrt 2 +sqrt 3)/2not in Bbb Q.$



                                    (iii). In the general case, for metric spacess it is a useful, widely used result that every member of $B|Y$ is a union of open $d$-balls of $Y,$ so the subspace topology $T|Y,$ as a subspace of $X,$ co-incides with the topology on $Y$ generated by the metric $d|_{Ytimes Y}.$






                                    share|cite|improve this answer























                                      up vote
                                      0
                                      down vote










                                      up vote
                                      0
                                      down vote









                                      With the metric $d(x,y)=|x-y|$ on $Bbb Q$ and the topology on $Bbb Q$ generated by $d$: For $qin K$ let $r(q)=min (q-sqrt 2,,sqrt 3 -q,).$ Let $K(q)={q'in Bbb Q: d(q',q)<r(q)}.$



                                      Then $K(q)$ is open in $Bbb Q$ and $qin K(q)subset K.$



                                      So $cup_{qin K}K(q)$ is open in $Bbb Q.$ And we have $K=cup_{qin K}
                                      ,{q}subset cup_{qin K},K(q)subset cup_{qin K},K=K,$
                                      so $K=cup_{qin K}K(q)$ is open in $Bbb Q.$



                                      An easily overlooked point about this Q:



                                      (i). Let $T$ be a topology on a set $X$ and let $Y subset X.$ The subspace topology $T|Y$ on $Y$ is defined as $T|Y={tcap Y:tin T}. $ If $B$ is a base (basis) for $T$ then $B|Y={bcap Y: bin B}$ is a base for $T|Y.$



                                      (ii).Suppose $T$ is generated by a metric $d$ on $X,$ so that the set $B$ of open $d$-balls of $X$ is a base for $T$. So $B|Y$ is a base for $T|Y.$



                                      BUT in general $B|Y$ may NOT be the set of open $d$-balls of $Y.$ An open $d$-ball of $Y$ is $B_d^Y(y,r)={y'in Y: d(y',y)<r},$ for some $yin Y, $ which does belong to $B|Y,$ but there may be other members of $B|Y.$



                                      For example in your Q, with $X=Bbb R$ and $Y=Bbb Q$ and $d(u,v)=|u-v|,$ the set $K$ belongs to $B|Y$ but is not an open ball of $Bbb Q$ because $(sqrt 2 +sqrt 3)/2not in Bbb Q.$



                                      (iii). In the general case, for metric spacess it is a useful, widely used result that every member of $B|Y$ is a union of open $d$-balls of $Y,$ so the subspace topology $T|Y,$ as a subspace of $X,$ co-incides with the topology on $Y$ generated by the metric $d|_{Ytimes Y}.$






                                      share|cite|improve this answer












                                      With the metric $d(x,y)=|x-y|$ on $Bbb Q$ and the topology on $Bbb Q$ generated by $d$: For $qin K$ let $r(q)=min (q-sqrt 2,,sqrt 3 -q,).$ Let $K(q)={q'in Bbb Q: d(q',q)<r(q)}.$



                                      Then $K(q)$ is open in $Bbb Q$ and $qin K(q)subset K.$



                                      So $cup_{qin K}K(q)$ is open in $Bbb Q.$ And we have $K=cup_{qin K}
                                      ,{q}subset cup_{qin K},K(q)subset cup_{qin K},K=K,$
                                      so $K=cup_{qin K}K(q)$ is open in $Bbb Q.$



                                      An easily overlooked point about this Q:



                                      (i). Let $T$ be a topology on a set $X$ and let $Y subset X.$ The subspace topology $T|Y$ on $Y$ is defined as $T|Y={tcap Y:tin T}. $ If $B$ is a base (basis) for $T$ then $B|Y={bcap Y: bin B}$ is a base for $T|Y.$



                                      (ii).Suppose $T$ is generated by a metric $d$ on $X,$ so that the set $B$ of open $d$-balls of $X$ is a base for $T$. So $B|Y$ is a base for $T|Y.$



                                      BUT in general $B|Y$ may NOT be the set of open $d$-balls of $Y.$ An open $d$-ball of $Y$ is $B_d^Y(y,r)={y'in Y: d(y',y)<r},$ for some $yin Y, $ which does belong to $B|Y,$ but there may be other members of $B|Y.$



                                      For example in your Q, with $X=Bbb R$ and $Y=Bbb Q$ and $d(u,v)=|u-v|,$ the set $K$ belongs to $B|Y$ but is not an open ball of $Bbb Q$ because $(sqrt 2 +sqrt 3)/2not in Bbb Q.$



                                      (iii). In the general case, for metric spacess it is a useful, widely used result that every member of $B|Y$ is a union of open $d$-balls of $Y,$ so the subspace topology $T|Y,$ as a subspace of $X,$ co-incides with the topology on $Y$ generated by the metric $d|_{Ytimes Y}.$







                                      share|cite|improve this answer












                                      share|cite|improve this answer



                                      share|cite|improve this answer










                                      answered 7 hours ago









                                      DanielWainfleet

                                      33.8k31647




                                      33.8k31647






















                                          up vote
                                          0
                                          down vote














                                          $K=[sqrt 2,sqrt 3]∩mathbb{Q}={qin mathbb{Q}|sqrt 2<q<sqrt 3}$ where$[sqrt 2,sqrt 3]$ is closed in R.



                                          From this I can conclude that K is not open subset of Q




                                          You're really not making it clear what your reasoning is. You seem to mostly just be restating the problem statement. Reading between the lines, your argument seems to be:




                                          1. K is an intersection between a closed set and a closed set.


                                          2. K is therefore closed.


                                          3. Therefore K is not open.



                                          The third statement is wrong; that a set is closed doesn't mean it's open. Presenting your argument explicitly helps others, and hopefully yourself, see what's wrong with it.



                                          If we have the open ball topology, then since $sqrt2 <q$, we know that there is "space" between $sqrt 2$ and $q$, and similarly for $sqrt3$. So given any $q$, we can take $epsilon_1$ to be half the distance between $sqrt2$ and $q$, $epsilon_2$ to be half the distance between $sqrt3$ and $q$, and $epsilon$ to be the minimum of $epsilon_1$ and $epsilon_2$. Then everything withing $epsilon$ of $q$ is in K, so $q$ is interior, and since $q$ is arbitrary, K is open.






                                          share|cite|improve this answer



























                                            up vote
                                            0
                                            down vote














                                            $K=[sqrt 2,sqrt 3]∩mathbb{Q}={qin mathbb{Q}|sqrt 2<q<sqrt 3}$ where$[sqrt 2,sqrt 3]$ is closed in R.



                                            From this I can conclude that K is not open subset of Q




                                            You're really not making it clear what your reasoning is. You seem to mostly just be restating the problem statement. Reading between the lines, your argument seems to be:




                                            1. K is an intersection between a closed set and a closed set.


                                            2. K is therefore closed.


                                            3. Therefore K is not open.



                                            The third statement is wrong; that a set is closed doesn't mean it's open. Presenting your argument explicitly helps others, and hopefully yourself, see what's wrong with it.



                                            If we have the open ball topology, then since $sqrt2 <q$, we know that there is "space" between $sqrt 2$ and $q$, and similarly for $sqrt3$. So given any $q$, we can take $epsilon_1$ to be half the distance between $sqrt2$ and $q$, $epsilon_2$ to be half the distance between $sqrt3$ and $q$, and $epsilon$ to be the minimum of $epsilon_1$ and $epsilon_2$. Then everything withing $epsilon$ of $q$ is in K, so $q$ is interior, and since $q$ is arbitrary, K is open.






                                            share|cite|improve this answer

























                                              up vote
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                                              $K=[sqrt 2,sqrt 3]∩mathbb{Q}={qin mathbb{Q}|sqrt 2<q<sqrt 3}$ where$[sqrt 2,sqrt 3]$ is closed in R.



                                              From this I can conclude that K is not open subset of Q




                                              You're really not making it clear what your reasoning is. You seem to mostly just be restating the problem statement. Reading between the lines, your argument seems to be:




                                              1. K is an intersection between a closed set and a closed set.


                                              2. K is therefore closed.


                                              3. Therefore K is not open.



                                              The third statement is wrong; that a set is closed doesn't mean it's open. Presenting your argument explicitly helps others, and hopefully yourself, see what's wrong with it.



                                              If we have the open ball topology, then since $sqrt2 <q$, we know that there is "space" between $sqrt 2$ and $q$, and similarly for $sqrt3$. So given any $q$, we can take $epsilon_1$ to be half the distance between $sqrt2$ and $q$, $epsilon_2$ to be half the distance between $sqrt3$ and $q$, and $epsilon$ to be the minimum of $epsilon_1$ and $epsilon_2$. Then everything withing $epsilon$ of $q$ is in K, so $q$ is interior, and since $q$ is arbitrary, K is open.






                                              share|cite|improve this answer















                                              $K=[sqrt 2,sqrt 3]∩mathbb{Q}={qin mathbb{Q}|sqrt 2<q<sqrt 3}$ where$[sqrt 2,sqrt 3]$ is closed in R.



                                              From this I can conclude that K is not open subset of Q




                                              You're really not making it clear what your reasoning is. You seem to mostly just be restating the problem statement. Reading between the lines, your argument seems to be:




                                              1. K is an intersection between a closed set and a closed set.


                                              2. K is therefore closed.


                                              3. Therefore K is not open.



                                              The third statement is wrong; that a set is closed doesn't mean it's open. Presenting your argument explicitly helps others, and hopefully yourself, see what's wrong with it.



                                              If we have the open ball topology, then since $sqrt2 <q$, we know that there is "space" between $sqrt 2$ and $q$, and similarly for $sqrt3$. So given any $q$, we can take $epsilon_1$ to be half the distance between $sqrt2$ and $q$, $epsilon_2$ to be half the distance between $sqrt3$ and $q$, and $epsilon$ to be the minimum of $epsilon_1$ and $epsilon_2$. Then everything withing $epsilon$ of $q$ is in K, so $q$ is interior, and since $q$ is arbitrary, K is open.







                                              share|cite|improve this answer














                                              share|cite|improve this answer



                                              share|cite|improve this answer








                                              edited 2 hours ago









                                              Yanko

                                              5,520723




                                              5,520723










                                              answered 7 hours ago









                                              Acccumulation

                                              6,6112616




                                              6,6112616






























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