Prove the cross of two open sets is also an open set.
$U$ 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
asked Nov 27 '18 at 2:20
kendal
337
add a comment |
$U$ 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
asked Nov 27 '18 at 2:20
kendal
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
add a comment |
$U$ 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
asked Nov 27 '18 at 2:20
kendal
337
$U$ 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
real-analysis metric-spaces
asked Nov 27 '18 at 2:20
kendal
337
asked Nov 27 '18 at 2:20
kendal
337
edited Nov 28 '18 at 2:51
asked Nov 27 '18 at 2:20
kendal
337
asked Nov 27 '18 at 2:20
kendal
337
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
add a comment |
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
add a comment |
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.
answered Nov 27 '18 at 2:39
LeB
986217
add a comment |
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1 Answer
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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.
answered Nov 27 '18 at 2:39
LeB
986217
add a comment |
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.
answered Nov 27 '18 at 2:39
LeB
986217
add a comment |
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.
answered Nov 27 '18 at 2:39
LeB
986217
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.
answered Nov 27 '18 at 2:39
LeB
986217
answered Nov 27 '18 at 2:39
LeB
986217
answered Nov 27 '18 at 2:39
LeB
986217
answered Nov 27 '18 at 2:39
LeB
986217
986217
add a comment |
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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