Closure & Closed Sets in metric space

In summary, the closure of a set A in a metric space is the set of all points that are either elements of A or limit points of A. It is closely related to the concept of convergence, as the limit of a sequence in a metric space belongs to the closure of the set containing the terms of the sequence. To determine if a set is closed, its closure must be equal to the set itself. In general, a set cannot be both open and closed in a metric space, but there are exceptions such as discrete and finite metric spaces. The closure and interior of a set in a metric space are complementary concepts, with the closure including all points "close" to the set and the interior including all points "inside" the set
  • #1
kingwinner
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Definition: Let F be a subset of a metric space X. F is called closed if whenever is a sequence in F which converges to a E X, then a E F. (i.e. F contains all limits of sequences in F) The closure of F is the set of all limits of sequences in F.

Claim 1: F is contained in the clousre of F.
Claim 2: The closure of F is closed.

How can we prove these formally?

For claim 1, I think we have to show x E F => x E cl(F).
For claim 2, we need to prove that cl(F) contains all limits of sequences in cl(F).
But how can we prove these?

I know these are supposed to be basic facts, but my book never gives examples of how to prove these from the definitions given above...and I have no clue...
Any help is appreciated!
 
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  • #2
For claim 1, yes, you want to show that if x is in F, then x is in the closure of F. For some element x in F, can you find a sequence in F that converges to x?

For claim 2, simply apply the definition of closure to what is being asked
 
  • #3
1) So we have to show that if x E F, then x is the limit of a sequence in F??
How can we show it? I can't figure it out...

2) The closure of F is the set of all limits of sequences in F.
But we need to prove that cl(F) contains all limits of sequences in cl(F). How?

Can somebody explain a little more, please?
Thanks!
 
  • #4
(1) Think about constant sequences.

(2) Clearly cl(F) = F u F' where F' is the set of limit points of F. Use this and think about how 'adding' F' to F changes the sequences in F - all you are doing is 'completing' the sequences in F, you are not introducing anything new, so there would be no more potential limit points to consider.
 
  • #5
(2) Let {c_n} be a convergent sequence in cl(F). We want to show that it has its limit in cl(F). We can find a sequence {a_n} of elements in F, such that for every n, d(a_n,c_n)<1/n (because c_n is the limit of some sequence in F). Therefore lim c_n = lim a_n in cl(F).
 

1. What is the definition of closure in a metric space?

The closure of a set A in a metric space is the set of all points in the space that are either elements of A or are limit points of A. In other words, the closure includes all the points that are "close" to the set A in the metric space. It can be denoted as cl(A) or A̅.

2. How is the closure of a set related to the concept of convergence in metric space?

The closure of a set A is closely related to the concept of convergence in metric space. A sequence of points in a metric space is said to converge to a point x if for any given distance ε, there exists a point in the sequence that is within ε distance from x. The closure of A can be seen as the set of all points that can be approached by the sequence from A. In other words, the limit of a sequence in a metric space belongs to the closure of the set containing the terms of the sequence.

3. How can you determine if a set is closed in a metric space?

A set is closed in a metric space if its closure is equal to the set itself. In other words, a set is closed if it contains all its limit points. This can be checked by taking the closure of the set and comparing it to the original set. If they are equal, then the set is closed. Another way to determine if a set is closed is to check if its complement is open in the metric space.

4. Can a set be both open and closed in a metric space?

In general, a set cannot be both open and closed in a metric space. However, in some specific cases, a set can be both open and closed. For example, in a discrete metric space, every set is both open and closed. In a finite metric space, a set with only one element is both open and closed.

5. What is the relationship between closure and interior of a set in a metric space?

The closure and interior of a set in a metric space are complementary concepts. The closure of a set A is the smallest closed set that contains A, while the interior of A is the largest open set contained in A. In other words, the closure includes all the points that are "close" to A, while the interior includes all the points that are "inside" A. It can be shown that the closure of A is equal to the complement of the interior of A in a metric space.

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