Closure of f(A): Is it a Closed Set?

In summary, the closure of any set is by definition a closed set. To prove that the closure of a set A is closed, we can use the definition of closure as the smallest closed set that contains A. If we consider the complement of the closure of A, we can show that it is open by showing that it does not contain any limit points of A. Therefore, the closure of A is a closed set.
  • #1
poutsos.A
102
1
Given that f is a function from R(=real Nos) to R continuous on R AND ,A any subset of R,IS THE closure of f(A) ,a closed set??
 
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  • #2


The closure of any set is by definition a closed set. I think you should rephrase your question.
 
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  • #3


Yes, you right thank you. But if we define a set to be closed if its complement is open,
how then we prove its closure to be a closed set??
 
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  • #4


What definition of "closure of A" are you using?
 
  • #5


poutsos.A said:
Yes, you right thank you. But if we define a set to be closed if its complement is open,
how then we prove its closure to be a closed set??

what? by definition the closure of a set A is the smallest closed set that contains A.
 
  • #6


I presume the OP had in the mind the definition that the closure of S is the union of S and the set of its limit points. In this case:

Denote by S' the closure of S. Then we wish to show that S' is closed. Suppose x is in the complement of S'. Then x is not in S and is not a limit point of S. So there is an open ball around x that doesn't intersect S. This open ball cannot contain any limit point of S since if y is inside it, then there is a smaller ball centered at y contained in the bigger - and so there is an open ball around y that doesn't intersect S, so y is not a limit point of S. It follows that the open ball around x does not intersect S'. Therefore the complement of S' is open; so S' is closed.
 

Related to Closure of f(A): Is it a Closed Set?

1. What is the definition of a closed set?

A closed set is a set of points in a topological space that includes all of its limit points, meaning that every sequence of points in the set that converges also converges to a point in the set.

2. How is the closure of a set defined?

The closure of a set is defined as the smallest closed set that contains all points in the original set. It is the union of the original set and all of its limit points.

3. What is the relationship between the closure of a set and its limit points?

The closure of a set contains all of its limit points, meaning that the set includes all points that a sequence can approach as it converges. However, not all limit points are necessarily in the closure of the set.

4. How can I determine if a set is closed?

A set is closed if it contains all of its limit points. One way to determine this is by checking if the set is equal to its closure. If the closure of the set is equal to the set itself, then the set is closed.

5. Can a set be both open and closed?

In general, a set cannot be both open and closed. However, in some cases, a set can be both open and closed if it is the empty set or the entire space. In other words, a set is either open or closed, but not both.

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