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Prove that $U={x in Xmid d(x,A)

Proving that if $X$ is a complete metric space and $Asubset X$ is nowhere dense in $X$, then there is an open set in $X$ disjoint with $A$.Show that $D ={ x + y mid x in (0,1) ,y in [1,2) }$ is open or closedProve the existence of disjoint open subsetsWhy are ${0}$ and ${1}$ open subsets of the discrete metric space ${0,1}$?Disjoint closed subsets are respectively included in disjoint open subsets in a metric spaceUnderstanding closed and open ballsNot open set looks like the disjoint union of countably many open intervals.Is every open set a disjoint union of open balls?Is any subset of $mathbb{Z}$ open?Separating two points with open sets whose closure is also disjoint

4

$begingroup$

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

share|cite|improve this question

edited Mar 14 at 19:47

Asaf Karagila♦

307k33438770

asked Mar 14 at 13:34

Sean ThrasherSean Thrasher

444

$endgroup$

add a comment | 

4

$begingroup$

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

share|cite|improve this question

edited Mar 14 at 19:47

Asaf Karagila♦

307k33438770

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

share|cite|improve this question

edited Mar 14 at 19:47

Asaf Karagila♦

307k33438770

asked Mar 14 at 13:34

Sean ThrasherSean Thrasher

444

$endgroup$

share|cite|improve this question

edited Mar 14 at 19:47

Asaf Karagila♦

307k33438770

asked Mar 14 at 13:34

Sean ThrasherSean Thrasher

444

share|cite|improve this question

edited Mar 14 at 19:47

Asaf Karagila♦

307k33438770

asked Mar 14 at 13:34

Sean ThrasherSean Thrasher

444

edited Mar 14 at 19:47

Asaf Karagila♦

307k33438770

edited Mar 14 at 19:47

Asaf Karagila♦

307k33438770

asked Mar 14 at 13:34

Sean ThrasherSean Thrasher

444

asked Mar 14 at 13:34

Sean ThrasherSean Thrasher

444

2 Answers
2

active

oldest

votes

10

$begingroup$

$f(x)=d(x,B)-d(x,A)$ is continuous and $U=f^{-1}(x:x>0)$

share|cite|improve this answer

answered Mar 14 at 13:40

Tsemo AristideTsemo Aristide

59.8k11446

$endgroup$

add a comment | 

5

$begingroup$

Your strategy will not work in an arbitrary metric space. Consider for example
$ X = mathbb R setminus{0}$ 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).

share|cite|improve this answer

edited Mar 14 at 13:49

answered Mar 14 at 13:40

Henning MakholmHenning Makholm

242k17308551

$endgroup$

add a comment | 

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10

$begingroup$

$f(x)=d(x,B)-d(x,A)$ is continuous and $U=f^{-1}(x:x>0)$

share|cite|improve this answer

answered Mar 14 at 13:40

Tsemo AristideTsemo Aristide

59.8k11446

$endgroup$

add a comment | 

10

$begingroup$

$f(x)=d(x,B)-d(x,A)$ is continuous and $U=f^{-1}(x:x>0)$

share|cite|improve this answer

answered Mar 14 at 13:40

Tsemo AristideTsemo Aristide

59.8k11446

$endgroup$

add a comment | 

$begingroup$

$f(x)=d(x,B)-d(x,A)$ is continuous and $U=f^{-1}(x:x>0)$

share|cite|improve this answer

answered Mar 14 at 13:40

Tsemo AristideTsemo Aristide

59.8k11446

$endgroup$

share|cite|improve this answer

answered Mar 14 at 13:40

Tsemo AristideTsemo Aristide

59.8k11446

answered Mar 14 at 13:40

Tsemo AristideTsemo Aristide

59.8k11446

answered Mar 14 at 13:40

Tsemo AristideTsemo Aristide

59.8k11446

5

$begingroup$

Your strategy will not work in an arbitrary metric space. Consider for example
$ X = mathbb R setminus{0}$ 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).

share|cite|improve this answer

edited Mar 14 at 13:49

answered Mar 14 at 13:40

Henning MakholmHenning Makholm

242k17308551

$endgroup$

add a comment | 

5

$begingroup$

Your strategy will not work in an arbitrary metric space. Consider for example
$ X = mathbb R setminus{0}$ 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).

share|cite|improve this answer

edited Mar 14 at 13:49

answered Mar 14 at 13:40

Henning MakholmHenning Makholm

242k17308551

$endgroup$

add a comment | 

$begingroup$

Your strategy will not work in an arbitrary metric space. Consider for example
$ X = mathbb R setminus{0}$ 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).

share|cite|improve this answer

edited Mar 14 at 13:49

answered Mar 14 at 13:40

Henning MakholmHenning Makholm

242k17308551

$endgroup$

share|cite|improve this answer

edited Mar 14 at 13:49

answered Mar 14 at 13:40

Henning MakholmHenning Makholm

242k17308551

answered Mar 14 at 13:40

Henning MakholmHenning Makholm

242k17308551

answered Mar 14 at 13:40

Henning MakholmHenning Makholm

242k17308551