Let S be the subset of R. For n>=1, let A_(n)=[(1)/(2n+1),(1)/(2n)] and S=\cup u_(n)=1^(\infty )A_(n). Find the following: \int (S),ext(S),R\S,bd(S),S^('), the isolated points of S, and cl(S).

Given that \( S \) is defined as a subset of \( \mathbb{R} \), let's break down the set \( S \) and calculate the required properties.

### Problem Setup

For \( n \geq 1 \), let:

\[
A_n = \left[\frac{1}{2n+1}, \frac{1}{2n}\right]
\]

The set \( S \) is defined as:

\[
S = \bigcup_{n=1}^{\infty} A_n
\]

This means that \( S \) is the union of the intervals \( A_n \).

### 1. The Interior of \( S \), \( \text{int}(S) \)

The interior of a set is the largest open set contained within \( S \). Each \( A_n \) is a closed interval, and since these intervals are disjoint, there is no open set contained entirely within \( S \). Therefore, the interior of \( S \) is empty:

\[
\text{int}(S) = \emptyset
\]

### 2. The Exterior of \( S \), \( \text{ext}(S) \)

The exterior of \( S \) consists of all points that are not in the closure of \( S \). Since \( S \) is made up of closed intervals that are isolated from each other, the exterior of \( S \) is everything in \( \mathbb{R} \) except the points in these intervals:

\[
\text{ext}(S) = \mathbb{R} \setminus S
\]

### 3. The Complement of \( S \), \( \mathbb{R} \setminus S \)

The complement of \( S \) is the set of all points in \( \mathbb{R} \) that are not in \( S \). Since \( S \) consists of a union of closed intervals, the complement will consist of all the points outside these intervals. This can be written as:

\[
\mathbb{R} \setminus S = \bigcup_{n=1}^{\infty} \left( \left( \frac{1}{2n}, \frac{1}{2n-1} \right) \cup (-\infty, 0) \cup (1, \infty) \right)
\]

### 4. The Boundary of \( S \), \( \text{bd}(S) \)

The boundary of a set is the set of points that can be approached both from within the set and from outside the set. For each interval \( A_n = \left[\frac{1}{2n+1}, \frac{1}{2n}\right] \), the boundary consists of the two endpoints \( \frac{1}{2n+1} \) and \( \frac{1}{2n} \). Therefore, the boundary of \( S \) is:

\[
\text{bd}(S) = \left\{ \frac{1}{2n+1}, \frac{1}{2n} \mid n \geq 1 \right\}
\]

### 5. The Derived Set \( S' \)

The derived set \( S' \) consists of the limit points of \( S \). A limit point of \( S \) is a point where every neighborhood around that point contains at least one point of \( S \) distinct from the point itself. In this case, each interval \( A_n \) consists of isolated points, meaning there are no accumulation points. Therefore, the derived set \( S' \) is empty:

\[
S' = \emptyset
\]

### 6. The Isolated Points of \( S \)

The isolated points of \( S \) are those points in \( S \) that do not have any other points of \( S \) arbitrarily close to them. Since \( S \) consists of disjoint intervals, every point in \( S \) is isolated. Therefore, the set of isolated points of \( S \) is:

\[
\text{Isolated Points of } S = S
\]

### 7. The Closure of \( S \), \( \text{cl}(S) \)

The closure of \( S \) is the smallest closed set containing \( S \), which consists of all the points in \( S \) together with the limit points of \( S \). Since there are no limit points, the closure of \( S \) is simply \( S \) itself:

\[
\text{cl}(S) = S
\]

### Summary of Results:

1. **Interior of \( S \):**
   \[
   \text{int}(S) = \emptyset
   \]
2. **Exterior of \( S \):**
   \[
   \text{ext}(S) = \mathbb{R} \setminus S
   \]
3. **Complement of \( S \):**
   \[
   \mathbb{R} \setminus S = \text{All points outside the intervals } A_n
   \]
4. **Boundary of \( S \):**
   \[
   \text{bd}(S) = \

Explore the properties of the set S defined as the union of closed intervals [1/(2n+1), 1/(2n)] for n ≥ 1 in real analysis and topology. Learn about the interior, exterior, complement, boundary, isolated points, and closure of S.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation