Prove or Disprove: Closure of Int(X)=X

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In summary, closure in mathematics refers to the property of a set where every limit point of the set is also a member of the set. "Int(X)" represents the interior of a set X, which is the largest open set contained within X. An example of closure is the set of all rational numbers between 0 and 1 (exclusive) being closed to the set of all real numbers between 0 and 1 (inclusive). Closure is important in mathematics as it allows for the definition and analysis of set properties and plays a crucial role in various mathematical theories.
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julia89
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Prove or disprove

Closure of the Interior of a closed set X is equal to X
so clos(intX)=X

I think it is true, but i don't know how to prove it

I thought that clos(int(X))=int(X)+bdy(int(X))=X


thanks,

julia
 
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  • #2
Think carefully. Consider the singleton set {p} containing only the single point p. In any metric topology, such a set is closed. What is its interior? What is the closure of that interior.
 
  • #3
What if the interior is non-empty?
 

Related to Prove or Disprove: Closure of Int(X)=X

1. What is the definition of "closure" in mathematics?

Closure in mathematics refers to the property of a set that every limit point of the set is also a member of the set. In other words, it is the smallest closed set that contains all the points in a given set.

2. What does "Int(X)" mean in the statement "Closure of Int(X)=X"?

"Int(X)" represents the interior of a set X, which is the largest open set contained within X. It includes all the points in X that are not boundary points.

3. Can you give an example to illustrate the concept of closure?

Yes, for example, let X be the set of all rational numbers between 0 and 1 (exclusive). The closure of X would be the set of all real numbers between 0 and 1 (inclusive), as every point in X is a limit point of X and is also a member of X.

4. How can we prove or disprove the statement "Closure of Int(X)=X"?

To prove this statement, we can show that every limit point of Int(X) is also a member of X, and vice versa. This would demonstrate that the closure of Int(X) is equal to X. To disprove the statement, we can find a counterexample where the closure of Int(X) is not equal to X.

5. Why is the concept of closure important in mathematics?

Closure is an essential concept in mathematics as it allows us to define and analyze the properties of sets. It also plays a crucial role in various mathematical theories, including topology, functional analysis, and algebraic geometry.

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