# Closed and Open sets in R (or 'clopen')

• Bleys
In summary, The conversation discusses the proofs of a theorem in mathematical analysis by Apostol. The theorem states that a set A, which is both open and closed, is equal to the whole set of real numbers. The conversation also explores the concepts of interior, exterior, and boundary points, and how they relate to open and closed sets. It is concluded that for a set to be both open and closed, it must have no boundary points.
Bleys
I'm sure this has been asked before, but the proofs I've seen use the fact R is connected or continuous functions is some way. I'm trying to prove it with the things that have only been presented in the book so far (Mathematical Analysis by Apostol).

So, let A be a subset of R which is both open and closed. Assume A is non-empty. I want to end up showing A = R.
Now A is open so by the Representation theorem for open sets in R, A is the union of a countable collection of disjoint component intervals of A. Denote these intervals by $I_{k} = (a_{k},b_{k})$
But both $a_{k}$ and $b_{k}$ are accumulation points of A, and so must belong to A (for A is closed).
I'm not really sure how to proceed, or if I'm even using the right path. Would I then say, since A is open there is an r>0 such that $(b_{k}-r,b_{k}+r)$ is in A. Then use the representation again on this, and reiterate to conclude the whole real line is in A?

A point, p, is an "interior point" of set A, in a metric space. if and only if there exist $\delta> 0$ such that the $\delta$ neighborhood of p ($\{y | d(p, y)< \delta$) is a subset of A.

A point, p, is an "exterior point" of set A if and only if it is an interior point of the complement of A.

A point, p, is a "boundary point" of set A if and only if it is neither an interior point nor an exterior point of A.

Clearly a set contains all of its interior points and none of its exterior points. A boundary point of set A may or may not be in A. Also the interior points of A are the exterior points of complement of A and vice-versa. A set and its complement have the same boundary points.

One can show that a set is open if and only if it contains none of its boundary points and closed if and only if it contains all of its boundary points. In order to be both open and closed, A set would have to contain all of its boundary points and none of its boundary points. That is possible if and only if the set has no boundary points. For the set of real numbers with the "usual" topology, on R and the empty set have no boundary points.

## 1. What is the definition of a closed set in R?

A closed set in R is a subset of real numbers that includes all of its limit points. This means that if a sequence of numbers in the set converges to a point outside of the set, that point must also be included in the set. In other words, the set contains all of its boundary points.

## 2. How is a closed set different from an open set in R?

An open set in R is a subset of real numbers that does not include any of its boundary points. This means that for any point in the set, there is a small interval around it that only contains points within the set. In contrast, a closed set includes all of its boundary points.

## 3. What is a "clopen" set in R?

A "clopen" set in R is a set that is both closed and open. This means that the set includes all of its boundary points, but also does not include any of its boundary points. In other words, the set is both closed and open at the same time.

## 4. Can a set in R be both open and closed at the same time?

No, a set in R cannot be both open and closed at the same time. This is because an open set does not include any of its boundary points, while a closed set includes all of its boundary points. Therefore, a set cannot have both of these properties simultaneously.

## 5. How are closed and open sets used in real life applications?

Closed and open sets are used in a variety of real life applications, such as in topology, analysis, and geometry. They are also important in understanding concepts such as continuity, convergence, and compactness. In engineering and physics, closed and open sets are used to model and analyze systems and phenomena. Additionally, in data analysis and statistics, closed and open sets are used to define intervals and ranges of values, as well as to determine the convergence of numerical methods.

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