https://r2.erweima.ai/mathbot/image/1726329969606oylmaz07.jpg

This image contains several theorems and exercises in topology, particularly related to subsets of the real line \( \mathbb{R} \), closed sets, derived sets, and interior sets. Let’s go through each statement and provide a sketch or an outline of the proof for each.

### 1. **Demostrar: Si \( A \) es un subconjunto finito de \( \mathbb{R} \), entonces el conjunto derivado \( A' \) de \( A \) es vacío, i.e., \( A' = \emptyset \).**
   - **Proof Outline**: 
     - A derived set \( A' \) is the set of all accumulation points of \( A \).
     - For a finite subset of \( \mathbb{R} \), every point is isolated (there's some neighborhood around each point that contains no other points of \( A \)).
     - Since no point can be an accumulation point, \( A' = \emptyset \).

### 2. **Demostrar: Todo subconjunto finito de \( \mathbb{R} \) es cerrado.**
   - **Proof Outline**:
     - A set is closed if it contains all of its accumulation points.
     - As established in the previous statement, a finite set has no accumulation points, so it vacuously contains all of them.
     - Therefore, any finite set in \( \mathbb{R} \) is closed.

### 3. **Demostrar: Si \( A \subset B \), entonces \( A' \subset B' \).**
   - **Proof Outline**:
     - If \( A \subset B \), any accumulation point of \( A \) must also be an accumulation point of \( B \), because an accumulation point \( x \in A' \) means any neighborhood of \( x \) contains infinitely many points of \( A \), and since \( A \subset B \), it must also contain points of \( B \).
     - Thus, \( A' \subset B' \).

### 4. **Demostrar: Un subconjunto \( B \) de \( \mathbb{R}^2 \) es cerrado si y solo si \( d(p, B) = 0 \) implica \( p \in B \), donde \( d(p, B) = \inf \{ d(p, q) : q \in B \} \).**
   - **Proof Outline**:
     - The distance \( d(p, B) \) represents the shortest distance from a point \( p \) to the set \( B \).
     - If \( B \) is closed, all points \( p \) that are arbitrarily close to \( B \) must actually lie in \( B \). This means if \( d(p, B) = 0 \), then \( p \) must be in \( B \).
     - Conversely, if every point that is arbitrarily close to \( B \) is in \( B \), then \( B \) contains all of its limit points, which means it is closed.

### 5. **Demostrar: \( A \cup A' \) es cerrada para cualquier conjunto \( A \).**
   - **Proof Outline**:
     - \( A' \) is the set of all accumulation points of \( A \).
     - The closure of \( A \), denoted \( \overline{A} \), is \( A \cup A' \).
     - Since the closure of a set is always closed, \( A \cup A' \) is closed.

### 6. **Demostrar: \( A \cup A' \) es el menor conjunto cerrado que contiene \( A \), i.e., si \( F \) es cerrado y \( A \subset F \subset A \cup A' \), entonces \( F = A \cup A' \).**
   - **Proof Outline**:
     - By definition, \( A \cup A' \) is the closure of \( A \), and the closure is the smallest closed set that contains \( A \).
     - Any closed set \( F \) that contains \( A \) must also contain \( A' \), because the points in \( A' \) are limit points of \( A \).
     - Therefore, the smallest such set is \( A \cup A' \).

### 7. **Demostrar: El conjunto de los puntos interiores de un conjunto \( A \), denotado por \( \text{int}(A) \), es un conjunto abierto.**
   - **Proof Outline**:
     - The interior of \( A \), \( \text{int}(A) \), consists of all points of \( A \) that have a neighborhood entirely contained in \( A \).
     - By definition, any point with a neighborhood contained in \( A \) must be an interior point, and the set of all such points is open.
     - Therefore, \( \text{int}(A) \) is an open set.

### 8. **Demostrar: El conjunto de los puntos interiores es el mayor conjunto abierto contenido en \( A \), es decir, si \( G \) es abierto y \( G \subset A \), entonces \( \text{int}(A) = G \).**
   - **Proof Outline**:
     - The interior is the largest open set contained in \( A \) because it includes all points that have neighborhoods entirely within \( A \).
     - If \( G \) is an open set contained in \( A \), then every point in \( G \) must also be in \( \text{int}(A) \), since they both share the same property of having neighborhoods fully contained within \( A \).
     - Hence, \( \text{int}(A) \) is the largest open set contained in \( A \).

### 9. **Demostrar: Los únicos subconjuntos de \( \mathbb{R} \) que son, al mismo tiempo, abiertos y cerrados son \( \emptyset \) y \( \mathbb{R} \).**
   - **Proof Outline**:
     - A set that is both open and closed is called a **clopen** set.
     - In \( \mathbb{R} \) (and in any connected space), the only clopen sets are the empty set \( \emptyset \) and the whole space \( \mathbb{R} \), because any non-trivial open set would disconnect the space.
     - Therefore, the only clopen sets in \( \mathbb{R} \) are \( \emptyset \) and \( \mathbb{R} \).

---

Let me know if you'd like to go deeper into any specific proof or if you have questions! Here are some related questions:

1. How is the concept of derived sets \( A' \) generalized in other topologies?
2. Can a set in \( \mathbb{R} \) be both finite and have non-empty accumulation points?
3. What happens if we consider subsets of spaces other than \( \mathbb{R} \), such as higher-dimensional Euclidean spaces or more abstract topological spaces?
4. Can we have non-empty sets in \( \mathbb{R} \) that are neither open nor closed? What are examples?
5. What are the consequences of a set being both open and closed in spaces that are not connected?

**Tip**: In topology, closures and interiors play critical roles in analyzing the structure of sets. The closure includes all limit points, while the interior focuses on points that "fit" fully inside the set.

This image contains several theorems and exercises in topology, particularly related to subsets of the real line ℝ, closed sets, derived sets, and interior sets. The user is asked to prove various properties of finite sets, derived sets, and other topological constructs.

Explore key topology exercises proving properties of finite sets, derived sets, and closed sets in the real line. Includes detailed explanations for derived sets, interiors, and theorems on closures. Suitable for undergraduate students studying topology or real analysis.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation