https://r2.erweima.ai/mathbot/image/1726331558988gfhkgs8g.jpg

Here is a breakdown of the mathematical exercises given in the image, along with steps to solve them:

### Problem 1:
**Statement:**
"Demostrar: Un subconjunto \( B \) de \( \mathbb{R}^n \) es cerrado si y solo si \( d(p, B) = 0 \) implica \( p \in B \), donde \( d(p, B) = \inf \{ d(p, q) : q \in B \}. \)**

**Translation:**
Prove that a subset \( B \subseteq \mathbb{R}^n \) is closed if and only if \( d(p, B) = 0 \) implies \( p \in B \), where \( d(p, B) = \inf \{ d(p, q) : q \in B \} \).

**Solution Outline:**
1. **Definition of closed set:** A set \( B \) is closed if it contains all its limit points. That is, if a sequence \( \{ p_n \} \subset B \) converges to some point \( p \in \mathbb{R}^n \), then \( p \in B \).
   
2. **Distance function:** \( d(p, B) = \inf \{ d(p, q) : q \in B \} \) gives the shortest distance between \( p \) and the set \( B \).

3. **Forward direction (if \( B \) is closed):**
   - If \( B \) is closed and \( d(p, B) = 0 \), it means that there is a sequence \( \{ p_n \} \subset B \) such that \( p_n \to p \). By the definition of a closed set, \( p \in B \).

4. **Reverse direction (if \( d(p, B) = 0 \) implies \( p \in B \)):**
   - We need to show that \( B \) is closed. Suppose there is a sequence \( \{ p_n \} \subset B \) that converges to \( p \in \mathbb{R}^n \). Since \( d(p_n, B) = 0 \), this implies that \( p \in B \), proving that \( B \) is closed.

### Problem 2:
**Statement:**
"Demostrar: \( A \cup A' \) es cerrada para cualquier conjunto \( A \)."

**Translation:**
Prove that \( A \cup A' \) is closed for any set \( A \), where \( A' \) denotes the closure of \( A \).

**Solution Outline:**
1. **Closure of a set \( A \):** The closure \( A' \) of a set \( A \) is defined as the smallest closed set containing \( A \). It includes all limit points of \( A \).

2. **Union of a set and its closure:** By definition, \( A' \) is a closed set. The union \( A \cup A' \) is equal to \( A' \) because all points of \( A \) are either in \( A \) or in \( A' \), and \( A' \) is the closure of \( A \), meaning \( A' \) already contains all the limit points of \( A \).

3. **Conclusion:** Since \( A' \) is closed, and \( A \cup A' = A' \), the union is also closed.

### Problem 3:
**Statement:**
"Demostrar: \( A \cup A' \) es el menor conjunto cerrado que contiene \( A \), i.e., si \( F \) es cerrado y \( A \subseteq F \subseteq A \cup A' \), entonces \( F = A \cup A' \)."

**Translation:**
Prove that \( A \cup A' \) is the smallest closed set containing \( A \), i.e., if \( F \) is closed and \( A \subseteq F \subseteq A \cup A' \), then \( F = A \cup A' \).

**Solution Outline:**
1. **Smallest closed set:** \( A' \) is the closure of \( A \), which is the smallest closed set containing \( A \). By definition, any closed set \( F \) that contains \( A \) must contain \( A' \).

2. **Subset relations:** If \( F \) is closed and satisfies \( A \subseteq F \subseteq A \cup A' \), it must include all points of \( A \) and all limit points of \( A \) (which are in \( A' \)).

3. **Conclusion:** Therefore, \( F = A \cup A' \) since any closed set that contains \( A \) must also contain all its limit points, and the smallest such set is \( A \cup A' \).

---

Would you like further clarifications or details on any of these proofs? Here are five follow-up questions:

1. How is the distance function \( d(p, B) \) formally defined in metric spaces?
2. Can you explain the concept of limit points in relation to closed sets?
3. How does the closure \( A' \) relate to the concept of convergence of sequences?
4. What is the role of the infimum in the definition of \( d(p, B) \)?
5. Why does the union of a set \( A \) with its closure equal its closure?

**Tip:** A set is closed if it contains all its limit points, and the closure of a set is the smallest closed set that contains the original set.

Demostrar: Un subconjunto B de R^n es cerrado sii d(p, B) = 0 implica p ∈ B, donde d(p, B) = inf { d(p, q) : q ∈ B }.
Demostrar: A ∪ A' es cerrada para cualquier conjunto A.
Demostrar: A ∪ A' es el menor conjunto cerrado que contiene A, i.e., si F es cerrado y A ⊂ F ⊂ A ∪ A', entonces F = A ∪ A'.

This page provides detailed solutions for proving properties of closed sets and their closures in metric spaces. It includes concepts like distance functions, limit points, and the smallest closed sets containing a given set. Suitable for advanced university mathematics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation