Determine interior and boundary of $Atimes B$












1












$begingroup$


Let $(X,mathcal{O}_X)$ and $(Y,mathcal{O}_Y)$ be topological spaces and $A,B$ subset of $X,Y$ respectively. I have to find the interior $(Atimes B)^circ$ and the boundary $partial(Atimes B)$. Furthermore, it is that the open sets are given by the product topology on $Atimes B$.



This sounds pretty easy, nevertheless I am not quite sure what exactly I should show. The interior and boundary both depend on the topology itself. Is this context to general to actually "show" something?






general-topology







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    I suspect it wants you to find the interior and boundary in terms of the interiors and boundaries of $A$ and $B$ in their respective topologies.
    $endgroup$
    – gj255
    Dec 14 '17 at 17:47






  • 2




    $begingroup$
    For instance, wouldn't it be nice if $(Atimes B)^circ=A^circtimes B^circ$ and $partial(Atimes B)=partial Atimespartial B$? Sadly, only one of these is true (which one?) The other requires some modification. But this is the kind of thing they're after.
    $endgroup$
    – Arthur
    Dec 14 '17 at 17:54












  • $begingroup$
    I could show the first relation. Not sure about the second one though...
    $endgroup$
    – EpsilonDelta
    Dec 14 '17 at 18:24






  • 1




    $begingroup$
    Draw a picture in the plane using open intervals for $A$ and $B$ to get an feel for the boundary case. Or search the site.
    $endgroup$
    – Henno Brandsma
    Dec 14 '17 at 23:01
















1












$begingroup$


Let $(X,mathcal{O}_X)$ and $(Y,mathcal{O}_Y)$ be topological spaces and $A,B$ subset of $X,Y$ respectively. I have to find the interior $(Atimes B)^circ$ and the boundary $partial(Atimes B)$. Furthermore, it is that the open sets are given by the product topology on $Atimes B$.



This sounds pretty easy, nevertheless I am not quite sure what exactly I should show. The interior and boundary both depend on the topology itself. Is this context to general to actually "show" something?






general-topology







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    I suspect it wants you to find the interior and boundary in terms of the interiors and boundaries of $A$ and $B$ in their respective topologies.
    $endgroup$
    – gj255
    Dec 14 '17 at 17:47






  • 2




    $begingroup$
    For instance, wouldn't it be nice if $(Atimes B)^circ=A^circtimes B^circ$ and $partial(Atimes B)=partial Atimespartial B$? Sadly, only one of these is true (which one?) The other requires some modification. But this is the kind of thing they're after.
    $endgroup$
    – Arthur
    Dec 14 '17 at 17:54












  • $begingroup$
    I could show the first relation. Not sure about the second one though...
    $endgroup$
    – EpsilonDelta
    Dec 14 '17 at 18:24






  • 1




    $begingroup$
    Draw a picture in the plane using open intervals for $A$ and $B$ to get an feel for the boundary case. Or search the site.
    $endgroup$
    – Henno Brandsma
    Dec 14 '17 at 23:01














1












1








1





$begingroup$


Let $(X,mathcal{O}_X)$ and $(Y,mathcal{O}_Y)$ be topological spaces and $A,B$ subset of $X,Y$ respectively. I have to find the interior $(Atimes B)^circ$ and the boundary $partial(Atimes B)$. Furthermore, it is that the open sets are given by the product topology on $Atimes B$.



This sounds pretty easy, nevertheless I am not quite sure what exactly I should show. The interior and boundary both depend on the topology itself. Is this context to general to actually "show" something?






general-topology







share|cite|improve this question









$endgroup$




Let $(X,mathcal{O}_X)$ and $(Y,mathcal{O}_Y)$ be topological spaces and $A,B$ subset of $X,Y$ respectively. I have to find the interior $(Atimes B)^circ$ and the boundary $partial(Atimes B)$. Furthermore, it is that the open sets are given by the product topology on $Atimes B$.



This sounds pretty easy, nevertheless I am not quite sure what exactly I should show. The interior and boundary both depend on the topology itself. Is this context to general to actually "show" something?






general-topology




general-topology






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 14 '17 at 17:39









EpsilonDeltaEpsilonDelta

6921615




6921615








  • 1




    $begingroup$
    I suspect it wants you to find the interior and boundary in terms of the interiors and boundaries of $A$ and $B$ in their respective topologies.
    $endgroup$
    – gj255
    Dec 14 '17 at 17:47






  • 2




    $begingroup$
    For instance, wouldn't it be nice if $(Atimes B)^circ=A^circtimes B^circ$ and $partial(Atimes B)=partial Atimespartial B$? Sadly, only one of these is true (which one?) The other requires some modification. But this is the kind of thing they're after.
    $endgroup$
    – Arthur
    Dec 14 '17 at 17:54












  • $begingroup$
    I could show the first relation. Not sure about the second one though...
    $endgroup$
    – EpsilonDelta
    Dec 14 '17 at 18:24






  • 1




    $begingroup$
    Draw a picture in the plane using open intervals for $A$ and $B$ to get an feel for the boundary case. Or search the site.
    $endgroup$
    – Henno Brandsma
    Dec 14 '17 at 23:01














  • 1




    $begingroup$
    I suspect it wants you to find the interior and boundary in terms of the interiors and boundaries of $A$ and $B$ in their respective topologies.
    $endgroup$
    – gj255
    Dec 14 '17 at 17:47






  • 2




    $begingroup$
    For instance, wouldn't it be nice if $(Atimes B)^circ=A^circtimes B^circ$ and $partial(Atimes B)=partial Atimespartial B$? Sadly, only one of these is true (which one?) The other requires some modification. But this is the kind of thing they're after.
    $endgroup$
    – Arthur
    Dec 14 '17 at 17:54












  • $begingroup$
    I could show the first relation. Not sure about the second one though...
    $endgroup$
    – EpsilonDelta
    Dec 14 '17 at 18:24






  • 1




    $begingroup$
    Draw a picture in the plane using open intervals for $A$ and $B$ to get an feel for the boundary case. Or search the site.
    $endgroup$
    – Henno Brandsma
    Dec 14 '17 at 23:01








1




1




$begingroup$
I suspect it wants you to find the interior and boundary in terms of the interiors and boundaries of $A$ and $B$ in their respective topologies.
$endgroup$
– gj255
Dec 14 '17 at 17:47




$begingroup$
I suspect it wants you to find the interior and boundary in terms of the interiors and boundaries of $A$ and $B$ in their respective topologies.
$endgroup$
– gj255
Dec 14 '17 at 17:47




2




2




$begingroup$
For instance, wouldn't it be nice if $(Atimes B)^circ=A^circtimes B^circ$ and $partial(Atimes B)=partial Atimespartial B$? Sadly, only one of these is true (which one?) The other requires some modification. But this is the kind of thing they're after.
$endgroup$
– Arthur
Dec 14 '17 at 17:54






$begingroup$
For instance, wouldn't it be nice if $(Atimes B)^circ=A^circtimes B^circ$ and $partial(Atimes B)=partial Atimespartial B$? Sadly, only one of these is true (which one?) The other requires some modification. But this is the kind of thing they're after.
$endgroup$
– Arthur
Dec 14 '17 at 17:54














$begingroup$
I could show the first relation. Not sure about the second one though...
$endgroup$
– EpsilonDelta
Dec 14 '17 at 18:24




$begingroup$
I could show the first relation. Not sure about the second one though...
$endgroup$
– EpsilonDelta
Dec 14 '17 at 18:24




1




1




$begingroup$
Draw a picture in the plane using open intervals for $A$ and $B$ to get an feel for the boundary case. Or search the site.
$endgroup$
– Henno Brandsma
Dec 14 '17 at 23:01




$begingroup$
Draw a picture in the plane using open intervals for $A$ and $B$ to get an feel for the boundary case. Or search the site.
$endgroup$
– Henno Brandsma
Dec 14 '17 at 23:01










1 Answer
1






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1












$begingroup$

(To remove the question from unanswered).



According to Exercise 2.3.B from [Eng], $(Atimes B)^circ=A^circtimes B^circ$ and $partial(Atimes B)=partial Atimes overline{B}cup overline{A}times partial B$. Prove this.



By Proposition 2.3.1 from [Eng], the set $A^circtimes B^circ$ is open, so $A^circtimes B^circsubset (Atimes B)^circ$. On the other hand, let $(x,y)in (Atimes B)^circ$ be any point. Then there exists an element $Utimes V$ of the canonical base at $Xtimes Y$ such that $(x,y)in Utimes Vsubset Atimes B$. Then $Usubset A$ and $Vsubset B$. Since $U$ and $V$ are open in $X$ and $Y$, respectively, we have $Usubset A^circ$ and $Vsubset B^circ$. Then $(x,y)in A^circtimes B^circ$.



By Proposition 2.3.3 from [Eng], $overline{Atimes B}=overline Atimes overline B$, so



$$partial(Atimes B)=$$ $$overline{Atimes B}setminus (Atimes B)^circ=$$
$$overline Atimes overline B setminus A^circtimes B^circ =$$ $$
((partial Acup A^circ)times (partial Bcup B^circ)) setminus A^circtimes B^circ=$$ $$partial Atimes partial Bcup partial Atimes B^circ cup A^circ times partial B=$$ $$(partial Atimes partial Bcup partial Atimes B^circ)cup
(partial Atimes partial Bcup A^circ times partial B)=$$ $$ partial Atimes overline{B}cup overline{A}times partial B.$$



References



[Eng] Ryszard Engelking, General Topology, 2nd ed., Heldermann, Berlin, 1989.



enter image description hereenter image description hereenter image description hereenter image description hereenter image description here






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    $begingroup$

    (To remove the question from unanswered).



    According to Exercise 2.3.B from [Eng], $(Atimes B)^circ=A^circtimes B^circ$ and $partial(Atimes B)=partial Atimes overline{B}cup overline{A}times partial B$. Prove this.



    By Proposition 2.3.1 from [Eng], the set $A^circtimes B^circ$ is open, so $A^circtimes B^circsubset (Atimes B)^circ$. On the other hand, let $(x,y)in (Atimes B)^circ$ be any point. Then there exists an element $Utimes V$ of the canonical base at $Xtimes Y$ such that $(x,y)in Utimes Vsubset Atimes B$. Then $Usubset A$ and $Vsubset B$. Since $U$ and $V$ are open in $X$ and $Y$, respectively, we have $Usubset A^circ$ and $Vsubset B^circ$. Then $(x,y)in A^circtimes B^circ$.



    By Proposition 2.3.3 from [Eng], $overline{Atimes B}=overline Atimes overline B$, so



    $$partial(Atimes B)=$$ $$overline{Atimes B}setminus (Atimes B)^circ=$$
    $$overline Atimes overline B setminus A^circtimes B^circ =$$ $$
    ((partial Acup A^circ)times (partial Bcup B^circ)) setminus A^circtimes B^circ=$$     $$partial Atimes partial Bcup partial Atimes B^circ cup A^circ times partial B=$$ $$(partial Atimes partial Bcup partial Atimes B^circ)cup
    (partial Atimes partial Bcup A^circ times partial B)=$$     $$ partial Atimes overline{B}cup overline{A}times partial B.$$



    References



    [Eng] Ryszard Engelking, General Topology, 2nd ed., Heldermann, Berlin, 1989.



    enter image description hereenter image description hereenter image description hereenter image description hereenter image description here






    share|cite|improve this answer









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      1












      $begingroup$

      (To remove the question from unanswered).



      According to Exercise 2.3.B from [Eng], $(Atimes B)^circ=A^circtimes B^circ$ and $partial(Atimes B)=partial Atimes overline{B}cup overline{A}times partial B$. Prove this.



      By Proposition 2.3.1 from [Eng], the set $A^circtimes B^circ$ is open, so $A^circtimes B^circsubset (Atimes B)^circ$. On the other hand, let $(x,y)in (Atimes B)^circ$ be any point. Then there exists an element $Utimes V$ of the canonical base at $Xtimes Y$ such that $(x,y)in Utimes Vsubset Atimes B$. Then $Usubset A$ and $Vsubset B$. Since $U$ and $V$ are open in $X$ and $Y$, respectively, we have $Usubset A^circ$ and $Vsubset B^circ$. Then $(x,y)in A^circtimes B^circ$.



      By Proposition 2.3.3 from [Eng], $overline{Atimes B}=overline Atimes overline B$, so



      $$partial(Atimes B)=$$ $$overline{Atimes B}setminus (Atimes B)^circ=$$
      $$overline Atimes overline B setminus A^circtimes B^circ =$$ $$
      ((partial Acup A^circ)times (partial Bcup B^circ)) setminus A^circtimes B^circ=$$       $$partial Atimes partial Bcup partial Atimes B^circ cup A^circ times partial B=$$ $$(partial Atimes partial Bcup partial Atimes B^circ)cup
      (partial Atimes partial Bcup A^circ times partial B)=$$       $$ partial Atimes overline{B}cup overline{A}times partial B.$$



      References



      [Eng] Ryszard Engelking, General Topology, 2nd ed., Heldermann, Berlin, 1989.



      enter image description hereenter image description hereenter image description hereenter image description hereenter image description here






      share|cite|improve this answer









      $endgroup$
















        1












        1








        1





        $begingroup$

        (To remove the question from unanswered).



        According to Exercise 2.3.B from [Eng], $(Atimes B)^circ=A^circtimes B^circ$ and $partial(Atimes B)=partial Atimes overline{B}cup overline{A}times partial B$. Prove this.



        By Proposition 2.3.1 from [Eng], the set $A^circtimes B^circ$ is open, so $A^circtimes B^circsubset (Atimes B)^circ$. On the other hand, let $(x,y)in (Atimes B)^circ$ be any point. Then there exists an element $Utimes V$ of the canonical base at $Xtimes Y$ such that $(x,y)in Utimes Vsubset Atimes B$. Then $Usubset A$ and $Vsubset B$. Since $U$ and $V$ are open in $X$ and $Y$, respectively, we have $Usubset A^circ$ and $Vsubset B^circ$. Then $(x,y)in A^circtimes B^circ$.



        By Proposition 2.3.3 from [Eng], $overline{Atimes B}=overline Atimes overline B$, so



        $$partial(Atimes B)=$$ $$overline{Atimes B}setminus (Atimes B)^circ=$$
        $$overline Atimes overline B setminus A^circtimes B^circ =$$ $$
        ((partial Acup A^circ)times (partial Bcup B^circ)) setminus A^circtimes B^circ=$$         $$partial Atimes partial Bcup partial Atimes B^circ cup A^circ times partial B=$$ $$(partial Atimes partial Bcup partial Atimes B^circ)cup
        (partial Atimes partial Bcup A^circ times partial B)=$$         $$ partial Atimes overline{B}cup overline{A}times partial B.$$



        References



        [Eng] Ryszard Engelking, General Topology, 2nd ed., Heldermann, Berlin, 1989.



        enter image description hereenter image description hereenter image description hereenter image description hereenter image description here






        share|cite|improve this answer









        $endgroup$



        (To remove the question from unanswered).



        According to Exercise 2.3.B from [Eng], $(Atimes B)^circ=A^circtimes B^circ$ and $partial(Atimes B)=partial Atimes overline{B}cup overline{A}times partial B$. Prove this.



        By Proposition 2.3.1 from [Eng], the set $A^circtimes B^circ$ is open, so $A^circtimes B^circsubset (Atimes B)^circ$. On the other hand, let $(x,y)in (Atimes B)^circ$ be any point. Then there exists an element $Utimes V$ of the canonical base at $Xtimes Y$ such that $(x,y)in Utimes Vsubset Atimes B$. Then $Usubset A$ and $Vsubset B$. Since $U$ and $V$ are open in $X$ and $Y$, respectively, we have $Usubset A^circ$ and $Vsubset B^circ$. Then $(x,y)in A^circtimes B^circ$.



        By Proposition 2.3.3 from [Eng], $overline{Atimes B}=overline Atimes overline B$, so



        $$partial(Atimes B)=$$ $$overline{Atimes B}setminus (Atimes B)^circ=$$
        $$overline Atimes overline B setminus A^circtimes B^circ =$$ $$
        ((partial Acup A^circ)times (partial Bcup B^circ)) setminus A^circtimes B^circ=$$         $$partial Atimes partial Bcup partial Atimes B^circ cup A^circ times partial B=$$ $$(partial Atimes partial Bcup partial Atimes B^circ)cup
        (partial Atimes partial Bcup A^circ times partial B)=$$         $$ partial Atimes overline{B}cup overline{A}times partial B.$$



        References



        [Eng] Ryszard Engelking, General Topology, 2nd ed., Heldermann, Berlin, 1989.



        enter image description hereenter image description hereenter image description hereenter image description hereenter image description here







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        answered Jan 4 at 2:38









        Alex RavskyAlex Ravsky

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