Closed set equivalence theorem

In summary: Therefore, there exists a point in Compliment(S) that is not in Closure(S).4. But, z is not that point. 5. Therefore, z is not in Compliment(S).6. Therefore, z is in Closure(S).7. Therefore, z is not in Compliment(S).8. So, z is in Compliment(S), and is the solution to the homework problem.In summary, if you have a set S and you want to find the closure of S (the set of all interior points), you need to first find a point in S that is not in the closure of S and then you need
  • #1
ssayan3
15
0

Homework Statement


Hi guys, this problem gave me some trouble before, but I'd like to know if I have it worked out now...

"If S = S[tex]\cup[/tex]BdyS, then S is closed (S[tex]_{compliment}[/tex] is open)



Homework Equations


S is equal to it's closure.


The Attempt at a Solution


1. Pick a point p in S[tex]^{compliment}[/tex].
2. For all points q[tex]^{n}[/tex] in S, let [tex]\delta[/tex] = min{|p-q[tex]^{n}[/tex]|}
3. Dfine B(p,[tex]\delta[/tex])
4. For any point in S[tex]^{compliment}[/tex], we can produce [tex]\delta[/tex]>0 such that B(p,[tex]\delta[/tex])[tex]\subset[/tex]S[tex]^{compliment}[/tex]. Therefore, all points in S[tex]^{compliment}[/tex] are interior points; therefore, S[tex]_{compliment}[/tex] is open, and S is closed.
 
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  • #2
I am having a hard time figuring out what you mean by "S[tex]\cup[/tex]BdyS"
 
  • #3
Oh! I'm sorry if I was unclear about something...

"S[tex]\cup[/tex]BdyS" refers to the union of the set S with its boundary, and is called Closure(S)

It can also be referred to as the union of the interior of S with the boundary of S.

(Someone tell me if I made an error!)
 
  • #4
ssayan3 said:

Homework Statement


Hi guys, this problem gave me some trouble before, but I'd like to know if I have it worked out now...

"If S = S[tex]\cup[/tex]BdyS, then S is closed (S[tex]_{compliment}[/tex] is open)



Homework Equations


S is equal to it's closure.


The Attempt at a Solution


1. Pick a point p in S[tex]^{compliment}[/tex].
2. For all points q[tex]^{n}[/tex] in S, let [tex]\delta[/tex] = min{|p-q[tex]^{n}[/tex]|}
How do you know there exist such a minimum? Not every set has a minimum. Every set of real numbers, bounded below, has an infimum (greatest lower bound). But then how do you know it is not 0?

3. Dfine B(p,[tex]\delta[/tex])
4. For any point in S[tex]^{compliment}[/tex], we can produce [tex]\delta[/tex]>0 such that B(p,[tex]\delta[/tex])[tex]\subset[/tex]S[tex]^{compliment}[/tex]. Therefore, all points in S[tex]^{compliment}[/tex] are interior points; therefore, S[tex]_{compliment}[/tex] is open, and S is closed.
 
  • #5
Argh, that makes it even worse for me because now I really have no idea how to finish this one.

What I have so far is:
1. Pick a point z in Compliment(S)
2.Then, z is not in S, and is not in Closure(S) by hypothesis
 

Related to Closed set equivalence theorem

1. What is the Closed Set Equivalence Theorem?

The Closed Set Equivalence Theorem is a mathematical theorem that states that a set is closed if and only if its complement is open. In other words, a set is closed if all of its limit points are contained within the set.

2. How is the Closed Set Equivalence Theorem used in mathematics?

The Closed Set Equivalence Theorem is used in various branches of mathematics, such as real analysis, topology, and functional analysis. It is a fundamental concept in these areas and is used to prove many other theorems.

3. What is the importance of the Closed Set Equivalence Theorem?

The Closed Set Equivalence Theorem is important because it helps us understand the relationship between closed and open sets. It also allows us to determine if a set is closed by looking at its complement, which can simplify mathematical proofs and calculations.

4. Can the Closed Set Equivalence Theorem be applied to infinite sets?

Yes, the Closed Set Equivalence Theorem can be applied to infinite sets. The theorem holds for both finite and infinite sets as long as the limit points of the set are contained within the set itself.

5. Are there any real-world applications of the Closed Set Equivalence Theorem?

Yes, the Closed Set Equivalence Theorem has real-world applications in physics, engineering, and computer science. It is used to analyze and solve problems related to continuity, convergence, and optimization, among others.

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