Is $(0,1]$ a closed or open set?Open/Closed Setsclosed and open set - set $S$ is open if and only if its complement is closed?How do I determine if a set is open or closed??Is if a set is not open and its complement not closed, is the set closed and its complement open?Is this set neither open nor closed or closed?Relatively open / Relatively closed setsShowing set is closed and other sets not closed.A closed subset of $[0,1]$ with no interiors and has measure exactly $1$(please check my work) Topology: interior,boundary,limit points, isolated points.Set $E = mathbb R setminus n in mathbb N $ is Open in $mathbb R$ ? Is it Closed in $mathbb R$?

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Is $(0,1]$ a closed or open set?


Open/Closed Setsclosed and open set - set $S$ is open if and only if its complement is closed?How do I determine if a set is open or closed??Is if a set is not open and its complement not closed, is the set closed and its complement open?Is this set neither open nor closed or closed?Relatively open / Relatively closed setsShowing set is closed and other sets not closed.A closed subset of $[0,1]$ with no interiors and has measure exactly $1$(please check my work) Topology: interior,boundary,limit points, isolated points.Set $E = mathbb R setminus n in mathbb N $ is Open in $mathbb R$ ? Is it Closed in $mathbb R$?













4












$begingroup$


Is $A=(0,1]$ a closed or open set?



I think it's not an open set because it is not a subset of its interior points. Mainly, $1in A$ but $1notin A^circ$.



If A is closed, then the complement is open. However, the complement $A^c$ is not open because it is not a subset of its interior points. Mainly, $0 in A^c$ but $0notin (A^c)^circ$










share|cite|improve this question











$endgroup$











  • $begingroup$
    $(0, 1]$ is a semi-open or semi-closed set.
    $endgroup$
    – Paras Khosla
    yesterday






  • 5




    $begingroup$
    Depends on the topology!
    $endgroup$
    – Jakobian
    yesterday






  • 9




    $begingroup$
    Unlike doors, subsets of topological spaces may be both open and closed, and they may be neither open nor closed. This is an example of how using every day words to name precise mathematical definitions can be misleading.
    $endgroup$
    – Simon
    yesterday







  • 1




    $begingroup$
    The answer to your question is no.
    $endgroup$
    – Robert Shore
    yesterday






  • 2




    $begingroup$
    @Simon doors can also be open and closed at the same time! You just need them to be adjoint to two different entrances at once.
    $endgroup$
    – John Dvorak
    yesterday















4












$begingroup$


Is $A=(0,1]$ a closed or open set?



I think it's not an open set because it is not a subset of its interior points. Mainly, $1in A$ but $1notin A^circ$.



If A is closed, then the complement is open. However, the complement $A^c$ is not open because it is not a subset of its interior points. Mainly, $0 in A^c$ but $0notin (A^c)^circ$










share|cite|improve this question











$endgroup$











  • $begingroup$
    $(0, 1]$ is a semi-open or semi-closed set.
    $endgroup$
    – Paras Khosla
    yesterday






  • 5




    $begingroup$
    Depends on the topology!
    $endgroup$
    – Jakobian
    yesterday






  • 9




    $begingroup$
    Unlike doors, subsets of topological spaces may be both open and closed, and they may be neither open nor closed. This is an example of how using every day words to name precise mathematical definitions can be misleading.
    $endgroup$
    – Simon
    yesterday







  • 1




    $begingroup$
    The answer to your question is no.
    $endgroup$
    – Robert Shore
    yesterday






  • 2




    $begingroup$
    @Simon doors can also be open and closed at the same time! You just need them to be adjoint to two different entrances at once.
    $endgroup$
    – John Dvorak
    yesterday













4












4








4


1



$begingroup$


Is $A=(0,1]$ a closed or open set?



I think it's not an open set because it is not a subset of its interior points. Mainly, $1in A$ but $1notin A^circ$.



If A is closed, then the complement is open. However, the complement $A^c$ is not open because it is not a subset of its interior points. Mainly, $0 in A^c$ but $0notin (A^c)^circ$










share|cite|improve this question











$endgroup$




Is $A=(0,1]$ a closed or open set?



I think it's not an open set because it is not a subset of its interior points. Mainly, $1in A$ but $1notin A^circ$.



If A is closed, then the complement is open. However, the complement $A^c$ is not open because it is not a subset of its interior points. Mainly, $0 in A^c$ but $0notin (A^c)^circ$







real-analysis






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 18 hours ago







Qwertford

















asked yesterday









QwertfordQwertford

323212




323212











  • $begingroup$
    $(0, 1]$ is a semi-open or semi-closed set.
    $endgroup$
    – Paras Khosla
    yesterday






  • 5




    $begingroup$
    Depends on the topology!
    $endgroup$
    – Jakobian
    yesterday






  • 9




    $begingroup$
    Unlike doors, subsets of topological spaces may be both open and closed, and they may be neither open nor closed. This is an example of how using every day words to name precise mathematical definitions can be misleading.
    $endgroup$
    – Simon
    yesterday







  • 1




    $begingroup$
    The answer to your question is no.
    $endgroup$
    – Robert Shore
    yesterday






  • 2




    $begingroup$
    @Simon doors can also be open and closed at the same time! You just need them to be adjoint to two different entrances at once.
    $endgroup$
    – John Dvorak
    yesterday
















  • $begingroup$
    $(0, 1]$ is a semi-open or semi-closed set.
    $endgroup$
    – Paras Khosla
    yesterday






  • 5




    $begingroup$
    Depends on the topology!
    $endgroup$
    – Jakobian
    yesterday






  • 9




    $begingroup$
    Unlike doors, subsets of topological spaces may be both open and closed, and they may be neither open nor closed. This is an example of how using every day words to name precise mathematical definitions can be misleading.
    $endgroup$
    – Simon
    yesterday







  • 1




    $begingroup$
    The answer to your question is no.
    $endgroup$
    – Robert Shore
    yesterday






  • 2




    $begingroup$
    @Simon doors can also be open and closed at the same time! You just need them to be adjoint to two different entrances at once.
    $endgroup$
    – John Dvorak
    yesterday















$begingroup$
$(0, 1]$ is a semi-open or semi-closed set.
$endgroup$
– Paras Khosla
yesterday




$begingroup$
$(0, 1]$ is a semi-open or semi-closed set.
$endgroup$
– Paras Khosla
yesterday




5




5




$begingroup$
Depends on the topology!
$endgroup$
– Jakobian
yesterday




$begingroup$
Depends on the topology!
$endgroup$
– Jakobian
yesterday




9




9




$begingroup$
Unlike doors, subsets of topological spaces may be both open and closed, and they may be neither open nor closed. This is an example of how using every day words to name precise mathematical definitions can be misleading.
$endgroup$
– Simon
yesterday





$begingroup$
Unlike doors, subsets of topological spaces may be both open and closed, and they may be neither open nor closed. This is an example of how using every day words to name precise mathematical definitions can be misleading.
$endgroup$
– Simon
yesterday





1




1




$begingroup$
The answer to your question is no.
$endgroup$
– Robert Shore
yesterday




$begingroup$
The answer to your question is no.
$endgroup$
– Robert Shore
yesterday




2




2




$begingroup$
@Simon doors can also be open and closed at the same time! You just need them to be adjoint to two different entrances at once.
$endgroup$
– John Dvorak
yesterday




$begingroup$
@Simon doors can also be open and closed at the same time! You just need them to be adjoint to two different entrances at once.
$endgroup$
– John Dvorak
yesterday










3 Answers
3






active

oldest

votes


















11












$begingroup$

Neither. It doesn't contain a neighbourhood of $1$, so it isn't open; nor is its complement, $(-infty,,0]cup (1,,infty)$, which doesn't contain a neighbourhood of $0$.






share|cite|improve this answer









$endgroup$












  • $begingroup$
    note that "its complement" is only what it is if we assume reals as the topology. $(0, 1]$ is open in itself when using the $D = |x-y|$ metric because the complement is the empty set.
    $endgroup$
    – John Dvorak
    yesterday







  • 2




    $begingroup$
    @JohnDvorak Absolutely true. These questions can have just about any answer with a "non-default" topology.
    $endgroup$
    – J.G.
    yesterday


















1












$begingroup$

It's important that you specify where you are considering the subset $A$.



If $A subset X$ with $ X = mathbbR$, J.G. is absolutely right in the usual topology of $mathbbR$.



If $A subset X$ with $ X = [0,1]$, $A^c = 0 $, which is closed in the usual topology, then $A = (0,1]$ is open.



In other words, it's important to specify in what topologic space $X$ you are considering $A$ as a subset. There are some stranger metrics which may define some different open sets where things can be different.






share|cite|improve this answer











$endgroup$




















    0












    $begingroup$

    A set is not a door.



    It is not the case that a set is either open or closed. It can also be neither or both.



    Indeed, your arguments correctly establish that $(0,1]$ is neither open nor closed as a subset of $mathbbR$ with the usual topology. The empty set $emptyset$ is always both open and closed, no matter what the ambient space is.






    share|cite|improve this answer









    $endgroup$












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      3 Answers
      3






      active

      oldest

      votes








      3 Answers
      3






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      11












      $begingroup$

      Neither. It doesn't contain a neighbourhood of $1$, so it isn't open; nor is its complement, $(-infty,,0]cup (1,,infty)$, which doesn't contain a neighbourhood of $0$.






      share|cite|improve this answer









      $endgroup$












      • $begingroup$
        note that "its complement" is only what it is if we assume reals as the topology. $(0, 1]$ is open in itself when using the $D = |x-y|$ metric because the complement is the empty set.
        $endgroup$
        – John Dvorak
        yesterday







      • 2




        $begingroup$
        @JohnDvorak Absolutely true. These questions can have just about any answer with a "non-default" topology.
        $endgroup$
        – J.G.
        yesterday















      11












      $begingroup$

      Neither. It doesn't contain a neighbourhood of $1$, so it isn't open; nor is its complement, $(-infty,,0]cup (1,,infty)$, which doesn't contain a neighbourhood of $0$.






      share|cite|improve this answer









      $endgroup$












      • $begingroup$
        note that "its complement" is only what it is if we assume reals as the topology. $(0, 1]$ is open in itself when using the $D = |x-y|$ metric because the complement is the empty set.
        $endgroup$
        – John Dvorak
        yesterday







      • 2




        $begingroup$
        @JohnDvorak Absolutely true. These questions can have just about any answer with a "non-default" topology.
        $endgroup$
        – J.G.
        yesterday













      11












      11








      11





      $begingroup$

      Neither. It doesn't contain a neighbourhood of $1$, so it isn't open; nor is its complement, $(-infty,,0]cup (1,,infty)$, which doesn't contain a neighbourhood of $0$.






      share|cite|improve this answer









      $endgroup$



      Neither. It doesn't contain a neighbourhood of $1$, so it isn't open; nor is its complement, $(-infty,,0]cup (1,,infty)$, which doesn't contain a neighbourhood of $0$.







      share|cite|improve this answer












      share|cite|improve this answer



      share|cite|improve this answer










      answered yesterday









      J.G.J.G.

      31.8k23250




      31.8k23250











      • $begingroup$
        note that "its complement" is only what it is if we assume reals as the topology. $(0, 1]$ is open in itself when using the $D = |x-y|$ metric because the complement is the empty set.
        $endgroup$
        – John Dvorak
        yesterday







      • 2




        $begingroup$
        @JohnDvorak Absolutely true. These questions can have just about any answer with a "non-default" topology.
        $endgroup$
        – J.G.
        yesterday
















      • $begingroup$
        note that "its complement" is only what it is if we assume reals as the topology. $(0, 1]$ is open in itself when using the $D = |x-y|$ metric because the complement is the empty set.
        $endgroup$
        – John Dvorak
        yesterday







      • 2




        $begingroup$
        @JohnDvorak Absolutely true. These questions can have just about any answer with a "non-default" topology.
        $endgroup$
        – J.G.
        yesterday















      $begingroup$
      note that "its complement" is only what it is if we assume reals as the topology. $(0, 1]$ is open in itself when using the $D = |x-y|$ metric because the complement is the empty set.
      $endgroup$
      – John Dvorak
      yesterday





      $begingroup$
      note that "its complement" is only what it is if we assume reals as the topology. $(0, 1]$ is open in itself when using the $D = |x-y|$ metric because the complement is the empty set.
      $endgroup$
      – John Dvorak
      yesterday





      2




      2




      $begingroup$
      @JohnDvorak Absolutely true. These questions can have just about any answer with a "non-default" topology.
      $endgroup$
      – J.G.
      yesterday




      $begingroup$
      @JohnDvorak Absolutely true. These questions can have just about any answer with a "non-default" topology.
      $endgroup$
      – J.G.
      yesterday











      1












      $begingroup$

      It's important that you specify where you are considering the subset $A$.



      If $A subset X$ with $ X = mathbbR$, J.G. is absolutely right in the usual topology of $mathbbR$.



      If $A subset X$ with $ X = [0,1]$, $A^c = 0 $, which is closed in the usual topology, then $A = (0,1]$ is open.



      In other words, it's important to specify in what topologic space $X$ you are considering $A$ as a subset. There are some stranger metrics which may define some different open sets where things can be different.






      share|cite|improve this answer











      $endgroup$

















        1












        $begingroup$

        It's important that you specify where you are considering the subset $A$.



        If $A subset X$ with $ X = mathbbR$, J.G. is absolutely right in the usual topology of $mathbbR$.



        If $A subset X$ with $ X = [0,1]$, $A^c = 0 $, which is closed in the usual topology, then $A = (0,1]$ is open.



        In other words, it's important to specify in what topologic space $X$ you are considering $A$ as a subset. There are some stranger metrics which may define some different open sets where things can be different.






        share|cite|improve this answer











        $endgroup$















          1












          1








          1





          $begingroup$

          It's important that you specify where you are considering the subset $A$.



          If $A subset X$ with $ X = mathbbR$, J.G. is absolutely right in the usual topology of $mathbbR$.



          If $A subset X$ with $ X = [0,1]$, $A^c = 0 $, which is closed in the usual topology, then $A = (0,1]$ is open.



          In other words, it's important to specify in what topologic space $X$ you are considering $A$ as a subset. There are some stranger metrics which may define some different open sets where things can be different.






          share|cite|improve this answer











          $endgroup$



          It's important that you specify where you are considering the subset $A$.



          If $A subset X$ with $ X = mathbbR$, J.G. is absolutely right in the usual topology of $mathbbR$.



          If $A subset X$ with $ X = [0,1]$, $A^c = 0 $, which is closed in the usual topology, then $A = (0,1]$ is open.



          In other words, it's important to specify in what topologic space $X$ you are considering $A$ as a subset. There are some stranger metrics which may define some different open sets where things can be different.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited yesterday









          Norrius

          1055




          1055










          answered yesterday









          521124521124

          78110




          78110





















              0












              $begingroup$

              A set is not a door.



              It is not the case that a set is either open or closed. It can also be neither or both.



              Indeed, your arguments correctly establish that $(0,1]$ is neither open nor closed as a subset of $mathbbR$ with the usual topology. The empty set $emptyset$ is always both open and closed, no matter what the ambient space is.






              share|cite|improve this answer









              $endgroup$

















                0












                $begingroup$

                A set is not a door.



                It is not the case that a set is either open or closed. It can also be neither or both.



                Indeed, your arguments correctly establish that $(0,1]$ is neither open nor closed as a subset of $mathbbR$ with the usual topology. The empty set $emptyset$ is always both open and closed, no matter what the ambient space is.






                share|cite|improve this answer









                $endgroup$















                  0












                  0








                  0





                  $begingroup$

                  A set is not a door.



                  It is not the case that a set is either open or closed. It can also be neither or both.



                  Indeed, your arguments correctly establish that $(0,1]$ is neither open nor closed as a subset of $mathbbR$ with the usual topology. The empty set $emptyset$ is always both open and closed, no matter what the ambient space is.






                  share|cite|improve this answer









                  $endgroup$



                  A set is not a door.



                  It is not the case that a set is either open or closed. It can also be neither or both.



                  Indeed, your arguments correctly establish that $(0,1]$ is neither open nor closed as a subset of $mathbbR$ with the usual topology. The empty set $emptyset$ is always both open and closed, no matter what the ambient space is.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered yesterday









                  ArnoArno

                  1,2381615




                  1,2381615



























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