# A set of real numbers whose interior is empty

• monkey372
In summary, the conversation discusses finding an example of a set of real numbers with an empty interior but a closure that includes all real numbers. It is suggested to think of a set that is "everywhere" on the real line, but not explicitly given as it would give away the answer. The attempt to find such an example is unsuccessful, leading to the possibility that no such set exists.
monkey372

## Homework Statement

Give an example of a set of real numbers whose interior is empty but whose closure is all of the real numbers if it exists. Otherwise, explain why such example cannot be true.

2. The attempt at a solution
For a set S ⊆ X, the closure of S is the intersection of all closed sets in X that contain A. I am having a lot of trouble thinking of an example and am beginning to think one does not exists but intuitively this does not make sense.

Last edited:
Try thinking of a set that's "everywhere" on the real line and compute the interiors of any such set you can think of. There is a more formal definition for this "everywhere"-ness that I'm alluding to. However, using that term directly would be handing you the answer.
e.g. [0,1] is certainly not "everywhere" on the real line.

## 1. What does it mean for a set of real numbers to have an empty interior?

For a set of real numbers to have an empty interior means that there are no points in the set that can be surrounded by an open interval. In other words, there are no points in the set that are not also boundary points.

## 2. Is it possible for a set of real numbers to have an empty interior?

Yes, it is possible for a set of real numbers to have an empty interior. This usually occurs when the set contains only boundary points or when the set is a closed interval with no interior points.

## 3. How is the interior of a set of real numbers determined?

The interior of a set of real numbers is determined by finding all the points in the set that are not boundary points. In other words, the interior of a set is the largest open set contained within the set.

## 4. What are some examples of sets of real numbers with an empty interior?

Examples of sets of real numbers with an empty interior include single points, closed intervals, and sets with only boundary points. For instance, the set [0,1] has an empty interior because it contains both its endpoints and no other points.

## 5. What is the significance of a set of real numbers having an empty interior in mathematics?

In mathematics, a set of real numbers with an empty interior can provide valuable information about the properties of the set. For example, sets with an empty interior are often used to define boundaries or to represent discrete values. Additionally, sets with an empty interior can help to distinguish between continuous and discrete functions.

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