Assume that $C$ is a compact set and $pin Xbackslash C $.

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$(X,d)$ is a metric space.Assume that $C$ is a compact set and $pin Xbackslash C $.

Construct two disjoint open sets $E_1$ and $E_2$ such that $pin E_1 $ and $Csubset E_2$.

I want to denote $d=d(p,C)$ which $d(p,C)$ is the distance between $p$ and $C$. And $E_1= B(p,fracd2)$ , $E_2=C$. I once thought it was figured out. But C is not open ...

So is there any brief construction?

general-topology

share|cite|improve this question

edited Jul 19 at 17:25

asked Jul 19 at 17:03

LOIS

997

• 
  
  
  

  
  

  
  
  
  
  
  

  
  C can be covered via a finite number open set. If $X$ is Hausdorff, then you can easily construct such $E_1$ and $E_2$. However, topological spaces (indicated by your tag) do not have to support metrics.
  – Imago
  Jul 19 at 17:10
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Is $X$ a metric space? There are some topologies in which this would not be possible.
  – EGoodman
  Jul 19 at 17:10
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @EGoodman It is a metric space.
  – LOIS
  Jul 19 at 17:26
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Then a hint would be to do it for two points first. However the answer below is most of what you need.
  – EGoodman
  Jul 19 at 18:48
  

  
  
  
  

add a comment | 

up vote
-1
down vote

favorite

$(X,d)$ is a metric space.Assume that $C$ is a compact set and $pin Xbackslash C $.

Construct two disjoint open sets $E_1$ and $E_2$ such that $pin E_1 $ and $Csubset E_2$.

I want to denote $d=d(p,C)$ which $d(p,C)$ is the distance between $p$ and $C$. And $E_1= B(p,fracd2)$ , $E_2=C$. I once thought it was figured out. But C is not open ...

So is there any brief construction?

general-topology

share|cite|improve this question

edited Jul 19 at 17:25

asked Jul 19 at 17:03

LOIS

997

• 
  
  
  

  
  

  
  
  
  
  
  

  
  C can be covered via a finite number open set. If $X$ is Hausdorff, then you can easily construct such $E_1$ and $E_2$. However, topological spaces (indicated by your tag) do not have to support metrics.
  – Imago
  Jul 19 at 17:10
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Is $X$ a metric space? There are some topologies in which this would not be possible.
  – EGoodman
  Jul 19 at 17:10
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @EGoodman It is a metric space.
  – LOIS
  Jul 19 at 17:26
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Then a hint would be to do it for two points first. However the answer below is most of what you need.
  – EGoodman
  Jul 19 at 18:48
  

  
  
  
  

add a comment | 

up vote
-1
down vote

favorite

up vote
-1
down vote

favorite

$(X,d)$ is a metric space.Assume that $C$ is a compact set and $pin Xbackslash C $.

Construct two disjoint open sets $E_1$ and $E_2$ such that $pin E_1 $ and $Csubset E_2$.

I want to denote $d=d(p,C)$ which $d(p,C)$ is the distance between $p$ and $C$. And $E_1= B(p,fracd2)$ , $E_2=C$. I once thought it was figured out. But C is not open ...

So is there any brief construction?

general-topology

share|cite|improve this question

edited Jul 19 at 17:25

asked Jul 19 at 17:03

LOIS

997

$(X,d)$ is a metric space.Assume that $C$ is a compact set and $pin Xbackslash C $.

Construct two disjoint open sets $E_1$ and $E_2$ such that $pin E_1 $ and $Csubset E_2$.

I want to denote $d=d(p,C)$ which $d(p,C)$ is the distance between $p$ and $C$. And $E_1= B(p,fracd2)$ , $E_2=C$. I once thought it was figured out. But C is not open ...

So is there any brief construction?

general-topology

share|cite|improve this question

edited Jul 19 at 17:25

asked Jul 19 at 17:03

LOIS

997

share|cite|improve this question

share|cite|improve this question

edited Jul 19 at 17:25

edited Jul 19 at 17:25

edited Jul 19 at 17:25

asked Jul 19 at 17:03

LOIS

997

asked Jul 19 at 17:03

LOIS

997

asked Jul 19 at 17:03

LOIS

997

997

• 
  
  
  

  
  

  
  
  
  
  
  

  
  C can be covered via a finite number open set. If $X$ is Hausdorff, then you can easily construct such $E_1$ and $E_2$. However, topological spaces (indicated by your tag) do not have to support metrics.
  – Imago
  Jul 19 at 17:10
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Is $X$ a metric space? There are some topologies in which this would not be possible.
  – EGoodman
  Jul 19 at 17:10
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @EGoodman It is a metric space.
  – LOIS
  Jul 19 at 17:26
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Then a hint would be to do it for two points first. However the answer below is most of what you need.
  – EGoodman
  Jul 19 at 18:48
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  C can be covered via a finite number open set. If $X$ is Hausdorff, then you can easily construct such $E_1$ and $E_2$. However, topological spaces (indicated by your tag) do not have to support metrics.
  – Imago
  Jul 19 at 17:10
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Is $X$ a metric space? There are some topologies in which this would not be possible.
  – EGoodman
  Jul 19 at 17:10
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @EGoodman It is a metric space.
  – LOIS
  Jul 19 at 17:26
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Then a hint would be to do it for two points first. However the answer below is most of what you need.
  – EGoodman
  Jul 19 at 18:48
  

  
  
  
  

C can be covered via a finite number open set. If $X$ is Hausdorff, then you can easily construct such $E_1$ and $E_2$. However, topological spaces (indicated by your tag) do not have to support metrics.
– Imago
Jul 19 at 17:10

C can be covered via a finite number open set. If $X$ is Hausdorff, then you can easily construct such $E_1$ and $E_2$. However, topological spaces (indicated by your tag) do not have to support metrics.
– Imago
Jul 19 at 17:10

Is $X$ a metric space? There are some topologies in which this would not be possible.
– EGoodman
Jul 19 at 17:10

Is $X$ a metric space? There are some topologies in which this would not be possible.
– EGoodman
Jul 19 at 17:10

@EGoodman It is a metric space.
– LOIS
Jul 19 at 17:26

@EGoodman It is a metric space.
– LOIS
Jul 19 at 17:26

Then a hint would be to do it for two points first. However the answer below is most of what you need.
– EGoodman
Jul 19 at 18:48

Then a hint would be to do it for two points first. However the answer below is most of what you need.
– EGoodman
Jul 19 at 18:48

add a comment | 

1 Answer
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up vote
1
down vote



accepted

For $xin C$, define $U_x = B(x,frac12d(x,p)), V_x=B(p,frac12d(x,p))$.

Then $U_x cap V_x = emptyset$, and letting $x$ vary in $C$ we get an open cover $U_x_x in C$.

Can you take it from here?

share|cite|improve this answer

answered Jul 19 at 18:16

Douglas Molin

27519

add a comment | 

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1 Answer
1

active

oldest

votes

1 Answer
1

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
1
down vote



accepted

For $xin C$, define $U_x = B(x,frac12d(x,p)), V_x=B(p,frac12d(x,p))$.

Then $U_x cap V_x = emptyset$, and letting $x$ vary in $C$ we get an open cover $U_x_x in C$.

Can you take it from here?

share|cite|improve this answer

answered Jul 19 at 18:16

Douglas Molin

27519

add a comment | 

up vote
1
down vote



accepted

For $xin C$, define $U_x = B(x,frac12d(x,p)), V_x=B(p,frac12d(x,p))$.

Then $U_x cap V_x = emptyset$, and letting $x$ vary in $C$ we get an open cover $U_x_x in C$.

Can you take it from here?

share|cite|improve this answer

answered Jul 19 at 18:16

Douglas Molin

27519

add a comment | 

up vote
1
down vote



accepted

up vote
1
down vote



accepted

For $xin C$, define $U_x = B(x,frac12d(x,p)), V_x=B(p,frac12d(x,p))$.

Then $U_x cap V_x = emptyset$, and letting $x$ vary in $C$ we get an open cover $U_x_x in C$.

Can you take it from here?

share|cite|improve this answer

answered Jul 19 at 18:16

Douglas Molin

27519

For $xin C$, define $U_x = B(x,frac12d(x,p)), V_x=B(p,frac12d(x,p))$.

Then $U_x cap V_x = emptyset$, and letting $x$ vary in $C$ we get an open cover $U_x_x in C$.

Can you take it from here?

share|cite|improve this answer

answered Jul 19 at 18:16

Douglas Molin

27519

share|cite|improve this answer

share|cite|improve this answer

answered Jul 19 at 18:16

Douglas Molin

27519

answered Jul 19 at 18:16

Douglas Molin

27519

answered Jul 19 at 18:16

Douglas Molin

27519

27519

add a comment | 

add a comment | 

 

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