Prove the cross of two open sets is also an open set.












0














$U$ and $V$ are open subsets of $mathbb{R}$. Show that $U×V = {(x,y)mid xin U, yin V }$ is an open subset of $mathbb{R}^{2}$.



I know the definition of an open subset is that $forall (x,y) in U×V, exists delta > 0$ such that $Bleft((x,y), deltaright) subset U×V$.



I just don't know how to use that in this case.










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  • We use a type of $LaTeX$ here called MathJax. There's an easy-to-find tutorial on it on the meta site.
    – Shaun
    Nov 27 '18 at 2:28












  • Please edit the question accordingly.
    – Shaun
    Nov 27 '18 at 2:29
















0














$U$ and $V$ are open subsets of $mathbb{R}$. Show that $U×V = {(x,y)mid xin U, yin V }$ is an open subset of $mathbb{R}^{2}$.



I know the definition of an open subset is that $forall (x,y) in U×V, exists delta > 0$ such that $Bleft((x,y), deltaright) subset U×V$.



I just don't know how to use that in this case.










share|cite|improve this question
























  • We use a type of $LaTeX$ here called MathJax. There's an easy-to-find tutorial on it on the meta site.
    – Shaun
    Nov 27 '18 at 2:28












  • Please edit the question accordingly.
    – Shaun
    Nov 27 '18 at 2:29














0












0








0







$U$ and $V$ are open subsets of $mathbb{R}$. Show that $U×V = {(x,y)mid xin U, yin V }$ is an open subset of $mathbb{R}^{2}$.



I know the definition of an open subset is that $forall (x,y) in U×V, exists delta > 0$ such that $Bleft((x,y), deltaright) subset U×V$.



I just don't know how to use that in this case.










share|cite|improve this question















$U$ and $V$ are open subsets of $mathbb{R}$. Show that $U×V = {(x,y)mid xin U, yin V }$ is an open subset of $mathbb{R}^{2}$.



I know the definition of an open subset is that $forall (x,y) in U×V, exists delta > 0$ such that $Bleft((x,y), deltaright) subset U×V$.



I just don't know how to use that in this case.







real-analysis metric-spaces






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edited Nov 28 '18 at 2:51

























asked Nov 27 '18 at 2:20









kendal

337




337












  • We use a type of $LaTeX$ here called MathJax. There's an easy-to-find tutorial on it on the meta site.
    – Shaun
    Nov 27 '18 at 2:28












  • Please edit the question accordingly.
    – Shaun
    Nov 27 '18 at 2:29


















  • We use a type of $LaTeX$ here called MathJax. There's an easy-to-find tutorial on it on the meta site.
    – Shaun
    Nov 27 '18 at 2:28












  • Please edit the question accordingly.
    – Shaun
    Nov 27 '18 at 2:29
















We use a type of $LaTeX$ here called MathJax. There's an easy-to-find tutorial on it on the meta site.
– Shaun
Nov 27 '18 at 2:28






We use a type of $LaTeX$ here called MathJax. There's an easy-to-find tutorial on it on the meta site.
– Shaun
Nov 27 '18 at 2:28














Please edit the question accordingly.
– Shaun
Nov 27 '18 at 2:29




Please edit the question accordingly.
– Shaun
Nov 27 '18 at 2:29










1 Answer
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Let $(x,y)in Utimes V$ be given.



Note that $xin U$ and $yin V$, so there exists $delta_x >0$ and $delta_y>0$ such that $$B(x,delta_x)subseteq U iff text{ and } B(y,delta_y)subseteq V.$$



Note that whenever $ain B(x,delta_x)$, we have $|x-a|<delta_x$. And similarly $|y-b|<delta_y$ whenever $bin B(y,delta_y)$.



Let $delta =min(delta_x,delta_y)$.



Now let $(a,b)in B((x,y),delta)$, then we have $sqrt{(x-a)^2+(y-b)^2}<delta$.



It implies that $|x-a|<deltaleq delta_x$ and that $|y-b|< delta leq delta_y$ which means that $ain U$ and $bin V$, so $(a,b)in Utimes V$.



Therefore, $B((x,y),delta)subseteq Utimes V$ and we are done.






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    1 Answer
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    Let $(x,y)in Utimes V$ be given.



    Note that $xin U$ and $yin V$, so there exists $delta_x >0$ and $delta_y>0$ such that $$B(x,delta_x)subseteq U iff text{ and } B(y,delta_y)subseteq V.$$



    Note that whenever $ain B(x,delta_x)$, we have $|x-a|<delta_x$. And similarly $|y-b|<delta_y$ whenever $bin B(y,delta_y)$.



    Let $delta =min(delta_x,delta_y)$.



    Now let $(a,b)in B((x,y),delta)$, then we have $sqrt{(x-a)^2+(y-b)^2}<delta$.



    It implies that $|x-a|<deltaleq delta_x$ and that $|y-b|< delta leq delta_y$ which means that $ain U$ and $bin V$, so $(a,b)in Utimes V$.



    Therefore, $B((x,y),delta)subseteq Utimes V$ and we are done.






    share|cite|improve this answer


























      2














      Let $(x,y)in Utimes V$ be given.



      Note that $xin U$ and $yin V$, so there exists $delta_x >0$ and $delta_y>0$ such that $$B(x,delta_x)subseteq U iff text{ and } B(y,delta_y)subseteq V.$$



      Note that whenever $ain B(x,delta_x)$, we have $|x-a|<delta_x$. And similarly $|y-b|<delta_y$ whenever $bin B(y,delta_y)$.



      Let $delta =min(delta_x,delta_y)$.



      Now let $(a,b)in B((x,y),delta)$, then we have $sqrt{(x-a)^2+(y-b)^2}<delta$.



      It implies that $|x-a|<deltaleq delta_x$ and that $|y-b|< delta leq delta_y$ which means that $ain U$ and $bin V$, so $(a,b)in Utimes V$.



      Therefore, $B((x,y),delta)subseteq Utimes V$ and we are done.






      share|cite|improve this answer
























        2












        2








        2






        Let $(x,y)in Utimes V$ be given.



        Note that $xin U$ and $yin V$, so there exists $delta_x >0$ and $delta_y>0$ such that $$B(x,delta_x)subseteq U iff text{ and } B(y,delta_y)subseteq V.$$



        Note that whenever $ain B(x,delta_x)$, we have $|x-a|<delta_x$. And similarly $|y-b|<delta_y$ whenever $bin B(y,delta_y)$.



        Let $delta =min(delta_x,delta_y)$.



        Now let $(a,b)in B((x,y),delta)$, then we have $sqrt{(x-a)^2+(y-b)^2}<delta$.



        It implies that $|x-a|<deltaleq delta_x$ and that $|y-b|< delta leq delta_y$ which means that $ain U$ and $bin V$, so $(a,b)in Utimes V$.



        Therefore, $B((x,y),delta)subseteq Utimes V$ and we are done.






        share|cite|improve this answer












        Let $(x,y)in Utimes V$ be given.



        Note that $xin U$ and $yin V$, so there exists $delta_x >0$ and $delta_y>0$ such that $$B(x,delta_x)subseteq U iff text{ and } B(y,delta_y)subseteq V.$$



        Note that whenever $ain B(x,delta_x)$, we have $|x-a|<delta_x$. And similarly $|y-b|<delta_y$ whenever $bin B(y,delta_y)$.



        Let $delta =min(delta_x,delta_y)$.



        Now let $(a,b)in B((x,y),delta)$, then we have $sqrt{(x-a)^2+(y-b)^2}<delta$.



        It implies that $|x-a|<deltaleq delta_x$ and that $|y-b|< delta leq delta_y$ which means that $ain U$ and $bin V$, so $(a,b)in Utimes V$.



        Therefore, $B((x,y),delta)subseteq Utimes V$ and we are done.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Nov 27 '18 at 2:39









        LeB

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