# Exterior and limit points: Quick Question

• Buri
In summary, the statement X\lim(A) = ext(A) is false, as the exterior of A is not equal to the set of limit points of A. This can be seen in the example of A= [0, 1]\cup \{2\} in R, where the isolated point 2 is not a limit point.
Buri
Is the following true?

X\lim(A) = ext(A), where A is a subset of a metric space X.

I think I've found a proof, but I don't feel very secure about it. So could someone just tell me if this is a correct assertion? What I'm trying to prove is that lim(A) is closed.

Thanks

No, this is false. The exterior of A can be expressed as X minus something, but that "something" is not the set of limit points of A. (As a hint, think about isolated points.)

Ahh really? Hmm I'm going to ahve to think about it more then...damnit

In R, consider $$A= [0, 1]\cup \{2\}$$.

Yup, those damn isolated points. Thanks!

## 1. What is an exterior point?

An exterior point is a point that lies outside of a set. In other words, it is a point that is not contained within the set.

## 2. How is an exterior point different from an interior point?

An interior point is a point that lies inside a set, while an exterior point lies outside of the set. Additionally, an interior point must be contained within the set, while an exterior point does not need to be contained within the set.

## 3. What is a limit point?

A limit point is a point that can be approximated by points within a set. In other words, if you draw a small circle around the limit point, there will always be points within the circle that are also in the set.

## 4. What is the relationship between exterior points and limit points?

Exterior points and limit points are essentially opposite concepts. Exterior points are points that are not contained within a set, while limit points are points that are contained within a set and can be approximated by other points in the set.

## 5. How are exterior and limit points used in math and science?

Exterior and limit points are important concepts in mathematical analysis and topology. They are used to define and study the properties of sets and to prove theorems in these fields.

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