Notation in point set Topology












1












$begingroup$


In $mathbb R setminus mathbb Q$ and $mathbb R /mathbb Q$, what do these ("$setminus$","$/$") symbols between the sets of real and rational numbers mean?






real-analysis general-topology notation







share|cite|improve this question











$endgroup$












  • $begingroup$
    Those are both versions of the "set difference", which is used probably determined by whichever notation is simpler to typeset. A/B= AB= the set of all elements of A that are NOT in B.
    $endgroup$
    – user247327
    12 hours ago






  • 7




    $begingroup$
    @user247327 $A/B$ does not denote the set theoretic difference!
    $endgroup$
    – Alex Kruckman
    12 hours ago
















1












$begingroup$


In $mathbb R setminus mathbb Q$ and $mathbb R /mathbb Q$, what do these ("$setminus$","$/$") symbols between the sets of real and rational numbers mean?






real-analysis general-topology notation







share|cite|improve this question











$endgroup$












  • $begingroup$
    Those are both versions of the "set difference", which is used probably determined by whichever notation is simpler to typeset. A/B= AB= the set of all elements of A that are NOT in B.
    $endgroup$
    – user247327
    12 hours ago






  • 7




    $begingroup$
    @user247327 $A/B$ does not denote the set theoretic difference!
    $endgroup$
    – Alex Kruckman
    12 hours ago














1












1








1





$begingroup$


In $mathbb R setminus mathbb Q$ and $mathbb R /mathbb Q$, what do these ("$setminus$","$/$") symbols between the sets of real and rational numbers mean?






real-analysis general-topology notation







share|cite|improve this question











$endgroup$




In $mathbb R setminus mathbb Q$ and $mathbb R /mathbb Q$, what do these ("$setminus$","$/$") symbols between the sets of real and rational numbers mean?






real-analysis general-topology notation




real-analysis general-topology notation






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 8 hours ago









GNUSupporter 8964民主女神 地下教會

12.8k72445




12.8k72445










asked 13 hours ago









user639820user639820

113




113












  • $begingroup$
    Those are both versions of the "set difference", which is used probably determined by whichever notation is simpler to typeset. A/B= AB= the set of all elements of A that are NOT in B.
    $endgroup$
    – user247327
    12 hours ago






  • 7




    $begingroup$
    @user247327 $A/B$ does not denote the set theoretic difference!
    $endgroup$
    – Alex Kruckman
    12 hours ago


















  • $begingroup$
    Those are both versions of the "set difference", which is used probably determined by whichever notation is simpler to typeset. A/B= AB= the set of all elements of A that are NOT in B.
    $endgroup$
    – user247327
    12 hours ago






  • 7




    $begingroup$
    @user247327 $A/B$ does not denote the set theoretic difference!
    $endgroup$
    – Alex Kruckman
    12 hours ago
















$begingroup$
Those are both versions of the "set difference", which is used probably determined by whichever notation is simpler to typeset. A/B= AB= the set of all elements of A that are NOT in B.
$endgroup$
– user247327
12 hours ago




$begingroup$
Those are both versions of the "set difference", which is used probably determined by whichever notation is simpler to typeset. A/B= AB= the set of all elements of A that are NOT in B.
$endgroup$
– user247327
12 hours ago




7




7




$begingroup$
@user247327 $A/B$ does not denote the set theoretic difference!
$endgroup$
– Alex Kruckman
12 hours ago




$begingroup$
@user247327 $A/B$ does not denote the set theoretic difference!
$endgroup$
– Alex Kruckman
12 hours ago










2 Answers
2






active

oldest

votes


















9












$begingroup$

Typically $mathbb{R}backslashmathbb{Q}$ is the set theoretic difference, i.e. the set of all irrationals in this case.



While $mathbb{R}/mathbb{Q}$ is the quotient group. It can also mean the result of collapsing a topological subspace to a point. Or the orbit space of $mathbb{Q}$ acting on $mathbb{R}$ and potentially other things. So it depends on the context. Either way it is some form of a quotient set.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Also $mathbb{R}/mathbb{Q}$ can denote $mathbb{R}$ viewed as a field extension of $mathbb{Q}$.
    $endgroup$
    – AlephNull
    8 hours ago



















3












$begingroup$

Depends on the context:



$mathbb{R} setminus mathbb{Q}$ is the set difference
between the reals and the rationals, so it equals the set of irrationals.



$mathbb{R}/mathbb{Q}$ can mean the quotient of the group of reals by its subgroup of the rationals, (which also gives a topological group) or it can denote a quotient space of the reals where we identify the subset of rationals to a single point.






share|cite|improve this answer









$endgroup$













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    2 Answers
    2






    active

    oldest

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    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    9












    $begingroup$

    Typically $mathbb{R}backslashmathbb{Q}$ is the set theoretic difference, i.e. the set of all irrationals in this case.



    While $mathbb{R}/mathbb{Q}$ is the quotient group. It can also mean the result of collapsing a topological subspace to a point. Or the orbit space of $mathbb{Q}$ acting on $mathbb{R}$ and potentially other things. So it depends on the context. Either way it is some form of a quotient set.






    share|cite|improve this answer











    $endgroup$













    • $begingroup$
      Also $mathbb{R}/mathbb{Q}$ can denote $mathbb{R}$ viewed as a field extension of $mathbb{Q}$.
      $endgroup$
      – AlephNull
      8 hours ago
















    9












    $begingroup$

    Typically $mathbb{R}backslashmathbb{Q}$ is the set theoretic difference, i.e. the set of all irrationals in this case.



    While $mathbb{R}/mathbb{Q}$ is the quotient group. It can also mean the result of collapsing a topological subspace to a point. Or the orbit space of $mathbb{Q}$ acting on $mathbb{R}$ and potentially other things. So it depends on the context. Either way it is some form of a quotient set.






    share|cite|improve this answer











    $endgroup$













    • $begingroup$
      Also $mathbb{R}/mathbb{Q}$ can denote $mathbb{R}$ viewed as a field extension of $mathbb{Q}$.
      $endgroup$
      – AlephNull
      8 hours ago














    9












    9








    9





    $begingroup$

    Typically $mathbb{R}backslashmathbb{Q}$ is the set theoretic difference, i.e. the set of all irrationals in this case.



    While $mathbb{R}/mathbb{Q}$ is the quotient group. It can also mean the result of collapsing a topological subspace to a point. Or the orbit space of $mathbb{Q}$ acting on $mathbb{R}$ and potentially other things. So it depends on the context. Either way it is some form of a quotient set.






    share|cite|improve this answer











    $endgroup$



    Typically $mathbb{R}backslashmathbb{Q}$ is the set theoretic difference, i.e. the set of all irrationals in this case.



    While $mathbb{R}/mathbb{Q}$ is the quotient group. It can also mean the result of collapsing a topological subspace to a point. Or the orbit space of $mathbb{Q}$ acting on $mathbb{R}$ and potentially other things. So it depends on the context. Either way it is some form of a quotient set.







    share|cite|improve this answer














    share|cite|improve this answer



    share|cite|improve this answer








    edited 12 hours ago

























    answered 12 hours ago









    freakishfreakish

    12.3k1630




    12.3k1630












    • $begingroup$
      Also $mathbb{R}/mathbb{Q}$ can denote $mathbb{R}$ viewed as a field extension of $mathbb{Q}$.
      $endgroup$
      – AlephNull
      8 hours ago


















    • $begingroup$
      Also $mathbb{R}/mathbb{Q}$ can denote $mathbb{R}$ viewed as a field extension of $mathbb{Q}$.
      $endgroup$
      – AlephNull
      8 hours ago
















    $begingroup$
    Also $mathbb{R}/mathbb{Q}$ can denote $mathbb{R}$ viewed as a field extension of $mathbb{Q}$.
    $endgroup$
    – AlephNull
    8 hours ago




    $begingroup$
    Also $mathbb{R}/mathbb{Q}$ can denote $mathbb{R}$ viewed as a field extension of $mathbb{Q}$.
    $endgroup$
    – AlephNull
    8 hours ago











    3












    $begingroup$

    Depends on the context:



    $mathbb{R} setminus mathbb{Q}$ is the set difference
    between the reals and the rationals, so it equals the set of irrationals.



    $mathbb{R}/mathbb{Q}$ can mean the quotient of the group of reals by its subgroup of the rationals, (which also gives a topological group) or it can denote a quotient space of the reals where we identify the subset of rationals to a single point.






    share|cite|improve this answer









    $endgroup$


















      3












      $begingroup$

      Depends on the context:



      $mathbb{R} setminus mathbb{Q}$ is the set difference
      between the reals and the rationals, so it equals the set of irrationals.



      $mathbb{R}/mathbb{Q}$ can mean the quotient of the group of reals by its subgroup of the rationals, (which also gives a topological group) or it can denote a quotient space of the reals where we identify the subset of rationals to a single point.






      share|cite|improve this answer









      $endgroup$
















        3












        3








        3





        $begingroup$

        Depends on the context:



        $mathbb{R} setminus mathbb{Q}$ is the set difference
        between the reals and the rationals, so it equals the set of irrationals.



        $mathbb{R}/mathbb{Q}$ can mean the quotient of the group of reals by its subgroup of the rationals, (which also gives a topological group) or it can denote a quotient space of the reals where we identify the subset of rationals to a single point.






        share|cite|improve this answer









        $endgroup$



        Depends on the context:



        $mathbb{R} setminus mathbb{Q}$ is the set difference
        between the reals and the rationals, so it equals the set of irrationals.



        $mathbb{R}/mathbb{Q}$ can mean the quotient of the group of reals by its subgroup of the rationals, (which also gives a topological group) or it can denote a quotient space of the reals where we identify the subset of rationals to a single point.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered 12 hours ago









        Henno BrandsmaHenno Brandsma

        109k347114




        109k347114






























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