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Prove that this set in the metric space is open
Open sets in a metric spaceOpen Balls in Metric Space.Why is the open ball in a discrete space with radius 2 the metric space itself?Why are $0$ and $1$ open subsets of the discrete metric space $0,1$?Open sets in Metric spacesEvery point of an open ball is a centre for the open ball.Is the empty set an open ball in a metric space?Defining a topology on a (non-metric) space…but what is the definition of 'open'?limit in open set in metric spaceIs every open set a disjoint union of open balls?
$begingroup$
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 5 hours ago
Mico
285210
asked 6 hours ago
Sean ThrasherSean Thrasher
334
$endgroup$
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 5 hours ago
Mico
285210
asked 6 hours ago
Sean ThrasherSean Thrasher
334
$endgroup$
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 5 hours ago
Mico
285210
asked 6 hours ago
Sean ThrasherSean Thrasher
334
$endgroup$
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 5 hours ago
Mico
285210
asked 6 hours ago
Sean ThrasherSean Thrasher
334
edited 5 hours ago
Mico
285210
asked 6 hours ago
Sean ThrasherSean Thrasher
334
edited 5 hours ago
Mico
285210
edited 5 hours ago
Mico
285210
edited 5 hours ago
Mico
285210
285210
asked 6 hours ago
Sean ThrasherSean Thrasher
334
asked 6 hours ago
Sean ThrasherSean Thrasher
334
asked 6 hours ago
Sean ThrasherSean Thrasher
334
334
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
$f(x)=d(x,B)-d(x,A)$ is continuous and $U=f^-1(x:x>0)$
answered 6 hours ago
Tsemo AristideTsemo Aristide
59.4k11446
$endgroup$
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 6 hours ago
Henning MakholmHenning Makholm
242k17308549
$endgroup$
add a comment |
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2 Answers
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2 Answers
2
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active
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votes
active
oldest
votes
$begingroup$
$f(x)=d(x,B)-d(x,A)$ is continuous and $U=f^-1(x:x>0)$
answered 6 hours ago
Tsemo AristideTsemo Aristide
59.4k11446
$endgroup$
add a comment |
$begingroup$
$f(x)=d(x,B)-d(x,A)$ is continuous and $U=f^-1(x:x>0)$
answered 6 hours ago
Tsemo AristideTsemo Aristide
59.4k11446
$endgroup$
add a comment |
$begingroup$
$f(x)=d(x,B)-d(x,A)$ is continuous and $U=f^-1(x:x>0)$
answered 6 hours ago
Tsemo AristideTsemo Aristide
59.4k11446
$endgroup$
$f(x)=d(x,B)-d(x,A)$ is continuous and $U=f^-1(x:x>0)$
answered 6 hours ago
Tsemo AristideTsemo Aristide
59.4k11446
answered 6 hours ago
Tsemo AristideTsemo Aristide
59.4k11446
answered 6 hours ago
Tsemo AristideTsemo Aristide
59.4k11446
answered 6 hours ago
Tsemo AristideTsemo Aristide
59.4k11446
59.4k11446
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 6 hours ago
Henning MakholmHenning Makholm
242k17308549
$endgroup$
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 6 hours ago
Henning MakholmHenning Makholm
242k17308549
$endgroup$
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 6 hours ago
Henning MakholmHenning Makholm
242k17308549
$endgroup$
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 6 hours ago
Henning MakholmHenning Makholm
242k17308549
edited 5 hours ago
answered 6 hours ago
Henning MakholmHenning Makholm
242k17308549
answered 6 hours ago
Henning MakholmHenning Makholm
242k17308549
answered 6 hours ago
Henning MakholmHenning Makholm
242k17308549
242k17308549
add a comment |
add a comment |
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