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Interior exterior and the boundary of a set in X with respect to cofinite topological space [closed]

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0

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The question is to find the interior and the exterior and the boundary of the set $A=[0,1)$ with respect to the cofinite topological space

general-topology

share|cite|improve this question

edited Mar 18 at 4:57

Henno Brandsma

114k348123

asked Mar 17 at 23:24

Rahmah issaRahmah issa

33

$endgroup$

closed as off-topic by Leucippus, Eevee Trainer, Parcly Taxel, mrtaurho, Cesareo Mar 18 at 8:27

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Leucippus, Eevee Trainer, Parcly Taxel, mrtaurho, Cesareo

If this question can be reworded to fit the rules in the help center, please edit the question.

add a comment | 

0

$begingroup$

The question is to find the interior and the exterior and the boundary of the set $A=[0,1)$ with respect to the cofinite topological space

general-topology

share|cite|improve this question

edited Mar 18 at 4:57

Henno Brandsma

114k348123

asked Mar 17 at 23:24

Rahmah issaRahmah issa

33

$endgroup$

closed as off-topic by Leucippus, Eevee Trainer, Parcly Taxel, mrtaurho, Cesareo Mar 18 at 8:27

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Leucippus, Eevee Trainer, Parcly Taxel, mrtaurho, Cesareo

If this question can be reworded to fit the rules in the help center, please edit the question.

add a comment | 

$begingroup$

The question is to find the interior and the exterior and the boundary of the set $A=[0,1)$ with respect to the cofinite topological space

general-topology

share|cite|improve this question

edited Mar 18 at 4:57

Henno Brandsma

114k348123

asked Mar 17 at 23:24

Rahmah issaRahmah issa

33

$endgroup$

share|cite|improve this question

edited Mar 18 at 4:57

Henno Brandsma

114k348123

asked Mar 17 at 23:24

Rahmah issaRahmah issa

33

share|cite|improve this question

edited Mar 18 at 4:57

Henno Brandsma

114k348123

asked Mar 17 at 23:24

Rahmah issaRahmah issa

33

edited Mar 18 at 4:57

Henno Brandsma

114k348123

edited Mar 18 at 4:57

Henno Brandsma

114k348123

asked Mar 17 at 23:24

Rahmah issaRahmah issa

33

asked Mar 17 at 23:24

Rahmah issaRahmah issa

33

2 Answers
2

active

oldest

votes

1

$begingroup$

It will depend on what $X$ (the total space) is. If $X=[0,1]$, then $A$ is itself open, being cofinite ($Xsetminus A=1$). Then the interior is $A$, and the exterior is $emptyset$ (as $A$ is dense).

If $X=mathbbR$, $A$ has empty interior, as it cannot contain a cofinite set and $A$ is dense because it's not finite nor $X$, making the exterior empty too.

share|cite|improve this answer

answered Mar 18 at 5:01

Henno BrandsmaHenno Brandsma

114k348123

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  X is the set of real numbers R thank you I understand
  $endgroup$
  – Rahmah issa
  Mar 18 at 21:39
  

  
  
  
  

add a comment | 

0

$begingroup$

Let A be a subset of a cofinite space S.

If A is cofinite, the interior of A is A,

the closure of A is S,

the boundary of A is S - A.

As the complement of A is finite, the exterior empty.

If A is not cofinite, then the interior is empty,

If A is infinite, then the closure of A is S,

. . the boundary of A is S and the exterior of A is empty.

If A is finite, then the closure of A is A,

. . the boundary of A is A and the exterior of A is S - A.

As you did not stipulate within what space [0,1) is a subset, I leave to you to apply the above.

Exercise. If [0,1) is a subset of [0,1] with the cofinite topology, what is the interior, closure, boundary and exterior of [0,1)?

share|cite|improve this answer

answered Mar 18 at 2:20

William ElliotWilliam Elliot

8,8582820

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thank you that is really helpful in the exercise I think that the boundary is X= [0,1] because it intersects the interior [0,1) and the exterior phi for all x correct me if I am wrong
  $endgroup$
  – Rahmah issa
  Mar 18 at 21:55
  

  
  
  
  

add a comment | 

1

$begingroup$

It will depend on what $X$ (the total space) is. If $X=[0,1]$, then $A$ is itself open, being cofinite ($Xsetminus A=1$). Then the interior is $A$, and the exterior is $emptyset$ (as $A$ is dense).

If $X=mathbbR$, $A$ has empty interior, as it cannot contain a cofinite set and $A$ is dense because it's not finite nor $X$, making the exterior empty too.

share|cite|improve this answer

answered Mar 18 at 5:01

Henno BrandsmaHenno Brandsma

114k348123

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  X is the set of real numbers R thank you I understand
  $endgroup$
  – Rahmah issa
  Mar 18 at 21:39
  

  
  
  
  

add a comment | 

1

$begingroup$

It will depend on what $X$ (the total space) is. If $X=[0,1]$, then $A$ is itself open, being cofinite ($Xsetminus A=1$). Then the interior is $A$, and the exterior is $emptyset$ (as $A$ is dense).

If $X=mathbbR$, $A$ has empty interior, as it cannot contain a cofinite set and $A$ is dense because it's not finite nor $X$, making the exterior empty too.

share|cite|improve this answer

answered Mar 18 at 5:01

Henno BrandsmaHenno Brandsma

114k348123

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  X is the set of real numbers R thank you I understand
  $endgroup$
  – Rahmah issa
  Mar 18 at 21:39
  

  
  
  
  

add a comment | 

$begingroup$

It will depend on what $X$ (the total space) is. If $X=[0,1]$, then $A$ is itself open, being cofinite ($Xsetminus A=1$). Then the interior is $A$, and the exterior is $emptyset$ (as $A$ is dense).

If $X=mathbbR$, $A$ has empty interior, as it cannot contain a cofinite set and $A$ is dense because it's not finite nor $X$, making the exterior empty too.

share|cite|improve this answer

answered Mar 18 at 5:01

Henno BrandsmaHenno Brandsma

114k348123

$endgroup$

share|cite|improve this answer

answered Mar 18 at 5:01

Henno BrandsmaHenno Brandsma

114k348123

answered Mar 18 at 5:01

Henno BrandsmaHenno Brandsma

114k348123

answered Mar 18 at 5:01

Henno BrandsmaHenno Brandsma

114k348123

0

$begingroup$

Let A be a subset of a cofinite space S.

If A is cofinite, the interior of A is A,

the closure of A is S,

the boundary of A is S - A.

As the complement of A is finite, the exterior empty.

If A is not cofinite, then the interior is empty,

If A is infinite, then the closure of A is S,

. . the boundary of A is S and the exterior of A is empty.

If A is finite, then the closure of A is A,

. . the boundary of A is A and the exterior of A is S - A.

As you did not stipulate within what space [0,1) is a subset, I leave to you to apply the above.

Exercise. If [0,1) is a subset of [0,1] with the cofinite topology, what is the interior, closure, boundary and exterior of [0,1)?

share|cite|improve this answer

answered Mar 18 at 2:20

William ElliotWilliam Elliot

8,8582820

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thank you that is really helpful in the exercise I think that the boundary is X= [0,1] because it intersects the interior [0,1) and the exterior phi for all x correct me if I am wrong
  $endgroup$
  – Rahmah issa
  Mar 18 at 21:55
  

  
  
  
  

add a comment | 

0

$begingroup$

Let A be a subset of a cofinite space S.

If A is cofinite, the interior of A is A,

the closure of A is S,

the boundary of A is S - A.

As the complement of A is finite, the exterior empty.

If A is not cofinite, then the interior is empty,

If A is infinite, then the closure of A is S,

. . the boundary of A is S and the exterior of A is empty.

If A is finite, then the closure of A is A,

. . the boundary of A is A and the exterior of A is S - A.

As you did not stipulate within what space [0,1) is a subset, I leave to you to apply the above.

Exercise. If [0,1) is a subset of [0,1] with the cofinite topology, what is the interior, closure, boundary and exterior of [0,1)?

share|cite|improve this answer

answered Mar 18 at 2:20

William ElliotWilliam Elliot

8,8582820

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thank you that is really helpful in the exercise I think that the boundary is X= [0,1] because it intersects the interior [0,1) and the exterior phi for all x correct me if I am wrong
  $endgroup$
  – Rahmah issa
  Mar 18 at 21:55
  

  
  
  
  

add a comment | 

$begingroup$

Let A be a subset of a cofinite space S.

If A is cofinite, the interior of A is A,

the closure of A is S,

the boundary of A is S - A.

As the complement of A is finite, the exterior empty.

If A is not cofinite, then the interior is empty,

If A is infinite, then the closure of A is S,

. . the boundary of A is S and the exterior of A is empty.

If A is finite, then the closure of A is A,

. . the boundary of A is A and the exterior of A is S - A.

As you did not stipulate within what space [0,1) is a subset, I leave to you to apply the above.

Exercise. If [0,1) is a subset of [0,1] with the cofinite topology, what is the interior, closure, boundary and exterior of [0,1)?

share|cite|improve this answer

answered Mar 18 at 2:20

William ElliotWilliam Elliot

8,8582820

$endgroup$

share|cite|improve this answer

answered Mar 18 at 2:20

William ElliotWilliam Elliot

8,8582820

answered Mar 18 at 2:20

William ElliotWilliam Elliot

8,8582820

answered Mar 18 at 2:20

William ElliotWilliam Elliot

8,8582820