How Do You Prove A Closure Equals A Union Boundary A?

In summary, the conversation is about proving that the closure of A is equal to the union of A and the boundary of A, and also proving that the boundary of A is equal to the intersection of the closure of A and the closure of the complement of A in R^n. The person is seeking help with writing this proof and asks if they can see the proof before it is answered.
  • #1
Virtate
6
0
How do you do this proof? Isn't it already obvious given the definition? I have no idea how to go about writing it down. :confused: If someone could help me with this, I would really appreciate it. Thanks :) Sorry I had to write it in such a messy way.

Define the closure of A as A closure = (x|for every open rectangle u containing x, intersection of u and A not equal to 0)

a) Show that A closure = A union bdA

b) Prove that bdA = intersection of A closure and (R^n-A) closure = bd(R^n-A)
 
Last edited:
Physics news on Phys.org
  • #2
Can we see your proof before we answer?
 

Related to How Do You Prove A Closure Equals A Union Boundary A?

1. What is the meaning of "A closure" in the statement "A closure = A union bdA"?

The closure of a set A, denoted by A-, is the smallest closed set that contains A. In other words, it is the union of A and all its limit points. In this particular statement, "A closure" refers to the closure of the set A.

2. How is the closure of a set related to its limit points?

The closure of a set A is the union of A and all its limit points. This means that any point in the closure of A is either a point in A or a limit point of A.

3. What does the "=" symbol mean in the statement "A closure = A union bdA"?

In this statement, "=" means that the closure of A is equal to the union of A and its boundary (bdA). This is known as the closure property, which states that the closure of a set is equal to the set itself plus its boundary.

4. How does the closure property relate to topology?

In topology, the closure property is a fundamental concept that is used to define closed sets. It states that a set is closed if and only if it is equal to its closure. This property is essential in understanding the behavior of sets in topological spaces.

5. What is the significance of A closure = A union bdA in mathematics?

This statement is a fundamental property of sets in mathematics. It helps us understand the relationship between a set and its closure, and how the boundary of a set can affect its closure. It is also used in various mathematical proofs and has applications in fields such as topology, analysis, and geometry.

Similar threads

Proving Closure and Identity for U(n)
    • Calculus and Beyond Homework Help
Replies
16
Views
4K
Proving Closure of A in Topological Space X
    • Calculus and Beyond Homework Help
Replies
12
Views
9K
Hausdorf space condition problem
    • Calculus and Beyond Homework Help
Replies
11
Views
1K
Closure of Open Ball in Normed Vector Spaces
    • Calculus and Beyond Homework Help
Replies
7
Views
2K
Finding 8 Relations on a Set of 3 Elements with the Same Symmetric Closure
    • Calculus and Beyond Homework Help
Replies
2
Views
1K
Linear subspaces and dimensions Proof
    • Calculus and Beyond Homework Help
Replies
8
Views
1K
I Closure & Interior as Dual Notions .... Proving Willard Theorem 3.11 ...
    • Topology and Analysis
Replies
2
Views
2K
The union of three subspaces of V is a subspace of V
    • Calculus and Beyond Homework Help
Replies
4
Views
2K
Proving closure and boundary points
    • Calculus and Beyond Homework Help
Replies
4
Views
2K
Union of a set and it's limit points
    • Calculus and Beyond Homework Help
Replies
3
Views
8K

Hot Threads

Recent Insights

Back
Top