# Finding the set of interior points, the closure, and an example

• splash_lover
In summary: So the closure of U will be larger than the closure of S. In summary, the set S=[0,1)U(1,2) has an interior set of (0,1)U(1,2) and a closure of [0,2]. An example of a set S of real numbers where U closure does not equal S closure is S=(0,3)U(5,6) and an example of a subset S of the interval [0,1] where S closure=[0,1] is S=(0,1/2)U(1/2,1).
splash_lover
Suppose that S=[0,1)U(1,2)

a) What is the set of interior points of S?

I thought it was (0,2)

b) Given that U is the set of interior points of S, evaluate U closure.

I thought that U closure=[0,2]

c) Give an example of a set S of real numbers such that if U is the set of interior points of S, then U closure DOES NOT equal S closure

This one I was not sure about, but here is my example:
S=(0,3)U(5,6) S closure=[0,3]U[5,6]
U=(0,6) U closure=[0,6]

d) Give an example of a subset S of the interval [0,1] such that S closure=[0,1].

I said if the subset S=(0,1/2)U(1/2,1) then S closure=[0,1]

Are my answers right for these? If not could you please explain what the answer is in detail?

Last edited:
If S=[0,1)U(1,2), as you wrote, is (1/2, 3/2) contained in S? (Note that 1 is an element of (1/2, 3/2)).

I think so, when I read the problem that's all it had was S=[0,1)U(1,2). So I am assuming (1/2,3/2) is contained in S.

OK, assume (1/2, 3/2) is an open subset of S. Since 1 is an element of (1/2, 3/2), can this statement be true?

no it can't because 1 is not included in S

Last edited:
OK, so, can (0, 2) be the interior of S?

no, (0,2) can't be the interior of S. So it would be (0,1)U(1,2)?

Yes. The closure of S you wrote down is correct. Note that a useful fact about closures is that a point is in the closure of a set if and only if every neighbourhood of that point intersects the set of interest. So, 1 is in the closure of S.

So the closure is [0,2].

Was the example i gave for part C correct?

Is it even possible to find an example for part c? I know the example I gave is wrong.

Can it be correct, considering all we said in between?

no, because some of the points in U (set of interior points)are not included in the original set S.

## 1. What are interior points?

Interior points are points within a set that are surrounded by other points in the set. In other words, they are points that do not lie on the boundary of the set.

## 2. How do you find the set of interior points for a given set?

The set of interior points can be found by determining all points in the set that have a neighborhood entirely contained within the set. This can be done by checking if each point in the set has a ball (a set of points within a certain distance from the given point) that is completely contained within the set.

## 3. What is the closure of a set?

The closure of a set is the set itself along with all of its limit points. A limit point is a point that can be approximated by points within the set. In other words, every neighborhood of a limit point contains points from the set.

## 4. How do you find the closure of a set?

The closure of a set can be found by taking the union of the set itself and its boundary. The boundary of a set is the set of all points that are neither interior points nor exterior points (points that are not in the set and are not limit points of the set).

## 5. Can you provide an example of finding the set of interior points and the closure of a set?

Sure, let's take the set A = {1, 2, 3, 4, 5, 6, 7}. The interior points of this set would be {2, 3, 4, 5, 6}, since each of these points has a neighborhood (for example, 2.5) that is completely contained within the set. The closure of this set would be {1, 2, 3, 4, 5, 6, 7}, since the boundary of the set is empty (there are no points that are neither interior nor exterior points).

• Calculus and Beyond Homework Help
Replies
3
Views
1K
• Calculus and Beyond Homework Help
Replies
1
Views
1K
• Calculus and Beyond Homework Help
Replies
7
Views
1K
• Calculus and Beyond Homework Help
Replies
4
Views
2K
• Calculus and Beyond Homework Help
Replies
6
Views
2K
• Calculus and Beyond Homework Help
Replies
29
Views
3K
• Calculus and Beyond Homework Help
Replies
5
Views
1K
• Calculus and Beyond Homework Help
Replies
3
Views
4K
• Calculus and Beyond Homework Help
Replies
8
Views
3K
• Calculus and Beyond Homework Help
Replies
6
Views
1K