Prove that $U=x in Xmid d(x,A)<d(x,B)$ is open when $A$ and $B$ are disjoint
Clash Royale CLAN TAG#URR8PPP
$begingroup$
In a metric space $(X,d)$, I have the set $$U=x in Xmid d(x,A)<d(x,B)$$
where $A$ and $B$ are disjoint subsets. I need to show that $U$ is open in $(X,d)$. I tried taking the radius of an open ball centre $xin U$ to be less than $d(x,C)$ where $C=xin Xmid d(x,A)=d(x,C)$ but I could not get anywhere. Is this the right strategy? I would really appreciate help, thank you!
metric-spaces
edited Mar 14 at 19:47
Asaf Karagila♦
308k33441775
asked Mar 14 at 13:34
Sean ThrasherSean Thrasher
444
$endgroup$
add a comment |
$begingroup$
In a metric space $(X,d)$, I have the set $$U=x in Xmid d(x,A)<d(x,B)$$
where $A$ and $B$ are disjoint subsets. I need to show that $U$ is open in $(X,d)$. I tried taking the radius of an open ball centre $xin U$ to be less than $d(x,C)$ where $C=xin Xmid d(x,A)=d(x,C)$ but I could not get anywhere. Is this the right strategy? I would really appreciate help, thank you!
metric-spaces
edited Mar 14 at 19:47
Asaf Karagila♦
308k33441775
asked Mar 14 at 13:34
Sean ThrasherSean Thrasher
444
$endgroup$
add a comment |
$begingroup$
In a metric space $(X,d)$, I have the set $$U=x in Xmid d(x,A)<d(x,B)$$
where $A$ and $B$ are disjoint subsets. I need to show that $U$ is open in $(X,d)$. I tried taking the radius of an open ball centre $xin U$ to be less than $d(x,C)$ where $C=xin Xmid d(x,A)=d(x,C)$ but I could not get anywhere. Is this the right strategy? I would really appreciate help, thank you!
metric-spaces
edited Mar 14 at 19:47
Asaf Karagila♦
308k33441775
asked Mar 14 at 13:34
Sean ThrasherSean Thrasher
444
$endgroup$
In a metric space $(X,d)$, I have the set $$U=x in Xmid d(x,A)<d(x,B)$$
where $A$ and $B$ are disjoint subsets. I need to show that $U$ is open in $(X,d)$. I tried taking the radius of an open ball centre $xin U$ to be less than $d(x,C)$ where $C=xin Xmid d(x,A)=d(x,C)$ but I could not get anywhere. Is this the right strategy? I would really appreciate help, thank you!
metric-spaces
metric-spaces
edited Mar 14 at 19:47
Asaf Karagila♦
308k33441775
asked Mar 14 at 13:34
Sean ThrasherSean Thrasher
444
edited Mar 14 at 19:47
Asaf Karagila♦
308k33441775
asked Mar 14 at 13:34
Sean ThrasherSean Thrasher
444
edited Mar 14 at 19:47
Asaf Karagila♦
308k33441775
edited Mar 14 at 19:47
Asaf Karagila♦
308k33441775
edited Mar 14 at 19:47
Asaf Karagila♦
308k33441775
308k33441775
asked Mar 14 at 13:34
Sean ThrasherSean Thrasher
444
asked Mar 14 at 13:34
Sean ThrasherSean Thrasher
444
asked Mar 14 at 13:34
Sean ThrasherSean Thrasher
444
444
add a comment |
add a comment |
2 Answers
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$begingroup$
$f(x)=d(x,B)-d(x,A)$ is continuous and $U=f^-1(x:x>0)$
answered Mar 14 at 13:40
Tsemo AristideTsemo Aristide
60.5k11446
$endgroup$
add a comment |
$begingroup$
Your strategy will not work in an arbitrary metric space. Consider for example
$ X = mathbb R setminus0$ with the usual distance and then $A=-1$, $B=1$. Then your $C$ is empty.
Instead, consider something like the ball around $xin U$ of radius $frac12(d(x,B)-d(x,A))$.
(Tsemo Aristide's suggestion is slicker than this, if you already know that $d(x,A)$ is a continuous function of $x$ and that preimages of open sets under continuous functions are open).
answered Mar 14 at 13:40
Henning MakholmHenning Makholm
243k17311555
$endgroup$
add a comment |
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2 Answers
2
active
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2 Answers
2
active
oldest
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$begingroup$
$f(x)=d(x,B)-d(x,A)$ is continuous and $U=f^-1(x:x>0)$
answered Mar 14 at 13:40
Tsemo AristideTsemo Aristide
60.5k11446
$endgroup$
add a comment |
$begingroup$
$f(x)=d(x,B)-d(x,A)$ is continuous and $U=f^-1(x:x>0)$
answered Mar 14 at 13:40
Tsemo AristideTsemo Aristide
60.5k11446
$endgroup$
add a comment |
$begingroup$
$f(x)=d(x,B)-d(x,A)$ is continuous and $U=f^-1(x:x>0)$
answered Mar 14 at 13:40
Tsemo AristideTsemo Aristide
60.5k11446
$endgroup$
$f(x)=d(x,B)-d(x,A)$ is continuous and $U=f^-1(x:x>0)$
answered Mar 14 at 13:40
Tsemo AristideTsemo Aristide
60.5k11446
answered Mar 14 at 13:40
Tsemo AristideTsemo Aristide
60.5k11446
answered Mar 14 at 13:40
Tsemo AristideTsemo Aristide
60.5k11446
answered Mar 14 at 13:40
Tsemo AristideTsemo Aristide
60.5k11446
60.5k11446
add a comment |
add a comment |
$begingroup$
Your strategy will not work in an arbitrary metric space. Consider for example
$ X = mathbb R setminus0$ with the usual distance and then $A=-1$, $B=1$. Then your $C$ is empty.
Instead, consider something like the ball around $xin U$ of radius $frac12(d(x,B)-d(x,A))$.
(Tsemo Aristide's suggestion is slicker than this, if you already know that $d(x,A)$ is a continuous function of $x$ and that preimages of open sets under continuous functions are open).
answered Mar 14 at 13:40
Henning MakholmHenning Makholm
243k17311555
$endgroup$
add a comment |
$begingroup$
Your strategy will not work in an arbitrary metric space. Consider for example
$ X = mathbb R setminus0$ with the usual distance and then $A=-1$, $B=1$. Then your $C$ is empty.
Instead, consider something like the ball around $xin U$ of radius $frac12(d(x,B)-d(x,A))$.
(Tsemo Aristide's suggestion is slicker than this, if you already know that $d(x,A)$ is a continuous function of $x$ and that preimages of open sets under continuous functions are open).
answered Mar 14 at 13:40
Henning MakholmHenning Makholm
243k17311555
$endgroup$
add a comment |
$begingroup$
Your strategy will not work in an arbitrary metric space. Consider for example
$ X = mathbb R setminus0$ with the usual distance and then $A=-1$, $B=1$. Then your $C$ is empty.
Instead, consider something like the ball around $xin U$ of radius $frac12(d(x,B)-d(x,A))$.
(Tsemo Aristide's suggestion is slicker than this, if you already know that $d(x,A)$ is a continuous function of $x$ and that preimages of open sets under continuous functions are open).
answered Mar 14 at 13:40
Henning MakholmHenning Makholm
243k17311555
$endgroup$
Your strategy will not work in an arbitrary metric space. Consider for example
$ X = mathbb R setminus0$ with the usual distance and then $A=-1$, $B=1$. Then your $C$ is empty.
Instead, consider something like the ball around $xin U$ of radius $frac12(d(x,B)-d(x,A))$.
(Tsemo Aristide's suggestion is slicker than this, if you already know that $d(x,A)$ is a continuous function of $x$ and that preimages of open sets under continuous functions are open).
answered Mar 14 at 13:40
Henning MakholmHenning Makholm
243k17311555
edited Mar 14 at 13:49
answered Mar 14 at 13:40
Henning MakholmHenning Makholm
243k17311555
answered Mar 14 at 13:40
Henning MakholmHenning Makholm
243k17311555
answered Mar 14 at 13:40
Henning MakholmHenning Makholm
243k17311555
243k17311555
add a comment |
add a comment |
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