# Cl(A) smallest closed set containing A.

• Buri
In summary, the conversation discusses a proof by a professor that Cl(A) (the closure of set A) is equal to the intersection of all closed sets containing A. The professor explains that since Cl(A) is one of the closed sets containing A, the intersection of all closed sets containing A is a subset of Cl(A). This is because when you intersect two sets, the resulting intersection will be a subset of both original sets. The conversation concludes with the realization that Cl(A) is indeed inside the intersection of all closed sets containing A.
Buri
My professor proved this result in class, but I don't understand the "simple" direction. He said that the above result is in another words proving that Cl(a) = intersection of all closed sets that contain A.

So he proved Cl(A) subset of intersection of all of the closed sets containing A. And intersection of closed sets containing A is a subset of Cl(A). However, I don't understand this last direction.

He simply says:

Cl(A) = int(A) U Bd(A); by definition.
=> Cl(A) is one of the closed sets containing A.
=> intersection of closed sets containing A is a subset of Cl(A).

That's what I just don't get. I understand that Cl(A) is closed and it contains A, but I just don't see how we can then say that the intersection of all closed sets containing A has Cl(A) in it. I know its going to be one of those I'll be intersecting all the other closed sets with, but that Cl(A) is actually INSIDE the intersection I just don't see.

I'm having a massive brain freeze so an explanation would be appreciated.

Thanks

If you intersect two sets, say A and B, then the intersection will be a subset of A. After all, it contains only elements in A (which satisfy a specific property, namely that they are also elements in B).
By the same argument, it will also be a subset of B.

Now suppose that you intersect Cl(A) with I, the intersection of all other closed sets containing A. Then Cl(A) intersected with I is a subset of both Cl(A) and of I. It is the former you are interested in here.

Ahhh thanks a lot. I get it. Stupid me just wasn't seeing it. Thanks a lot!

## 1. What does "Cl(A)" stand for in the phrase "smallest closed set containing A"?

"Cl(A)" stands for the closure of set A, which is the smallest closed set that contains all the points in A.

## 2. Why is the concept of "smallest closed set containing A" important in mathematics?

This concept is important because it allows us to define and study topological spaces, which are fundamental structures in mathematics. It also helps us to understand the behavior and properties of sets and their elements.

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

The closure of a set A contains all the limit points of A. In other words, every point in the closure of A can be approached by a sequence of points in A. This makes the closure an important tool in studying the convergence of sequences in topological spaces.

## 4. Can the closure of a set be empty?

Yes, the closure of a set can be empty if the set itself is empty. However, if the set contains at least one point, then its closure will always be non-empty.

## 5. How is the closure of a set different from its interior?

The closure of a set contains all the points in the set as well as its limit points, whereas the interior of a set only contains points that are not limit points. In other words, the interior is the largest open set contained in a given set, while the closure is the smallest closed set containing the same set.

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