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The image contains three mathematical propositions related to topology, specifically about open and closed sets in the context of the real numbers \( \mathbb{R} \). I'll walk through each one with explanations and proofs:

### 1. Proposition 1: 
**Statement:**  
"Demostrar: El conjunto de los puntos interiores de un conjunto \( A \), denotado por \( \text{int}(A) \), es un conjunto abierto."  
(*Prove that the set of interior points of a set \( A \), denoted by \( \text{int}(A) \), is an open set.*)

**Proof:**  
The interior of a set \( A \), denoted by \( \text{int}(A) \), is defined as the set of all points in \( A \) such that there exists an open neighborhood around each point, which is completely contained within \( A \). By this definition, every point in \( \text{int}(A) \) is by construction surrounded by an open neighborhood entirely contained in \( A \). Thus, \( \text{int}(A) \) itself must be open because any open set in \( \mathbb{R} \) is defined by the property that for each of its points, there exists an open neighborhood around it.

Therefore, \( \text{int}(A) \) is an open set.

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### 2. Proposition 2:
**Statement:**  
"Demostrar: El conjunto de los puntos interiores es el mayor conjunto abierto contenido en \( A \), es decir, si \( G \) es abierto y \( \text{int}(A) \subset G \subset A \), entonces \( \text{int}(A) = G \)."  
(*Prove that the interior of a set is the largest open set contained in \( A \). That is, if \( G \) is open and \( \text{int}(A) \subseteq G \subseteq A \), then \( \text{int}(A) = G \).*)

**Proof:**  
Let \( G \) be an open set such that \( \text{int}(A) \subset G \subset A \). Since \( \text{int}(A) \) is by definition the largest set of interior points where each point has an open neighborhood entirely contained in \( A \), any open set \( G \) contained in \( A \) cannot contain any points outside \( \text{int}(A) \). Otherwise, there would be points in \( G \) that are not interior points of \( A \), contradicting the maximality of \( \text{int}(A) \).

Thus, no such larger open set \( G \) can exist, and \( G = \text{int}(A) \). Therefore, the interior of a set \( A \) is indeed the largest open set contained within \( A \).

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### 3. Proposition 3:
**Statement:**  
"Demostrar: Los únicos subconjuntos de \( \mathbb{R} \) que son, al mismo tiempo, abiertos y cerrados son \( \varnothing \) y \( \mathbb{R} \)."  
(*Prove that the only subsets of \( \mathbb{R} \) that are both open and closed are \( \varnothing \) and \( \mathbb{R} \).*)

**Proof:**  
In topology, a set that is both open and closed is called "clopen." To prove this, let's assume \( A \subseteq \mathbb{R} \) is both open and closed.

1. **If \( A = \varnothing \) or \( A = \mathbb{R} \):**  
   Both the empty set \( \varnothing \) and the entire set \( \mathbb{R} \) are trivially both open and closed. The empty set has no points, so it satisfies the definitions of both openness and closedness. Similarly, \( \mathbb{R} \) contains all its limit points (closed) and for any point in \( \mathbb{R} \), there exists an open neighborhood (open).

2. **For any other non-trivial set \( A \subsetneq \mathbb{R} \):**  
   Consider the real line \( \mathbb{R} \) as a connected space, which means it cannot be divided into two disjoint non-empty open sets. If \( A \) is a proper subset of \( \mathbb{R} \) and is both open and closed, it would contradict the connectedness of \( \mathbb{R} \), as this would partition \( \mathbb{R} \) into two non-empty disjoint open sets, \( A \) and \( \mathbb{R} \setminus A \).

Thus, the only subsets of \( \mathbb{R} \) that can be both open and closed are \( \varnothing \) and \( \mathbb{R} \).

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Would you like further details on any of the proofs or concepts?  

Here are 5 related questions to expand on this:
1. What is the difference between an open set and a closed set in topology?
2. How do connected spaces play a role in the third proof?
3. Can there be other clopen sets in spaces other than \( \mathbb{R} \)?
4. What is the definition of a neighborhood in topology?
5. How does the concept of interior differ from closure in topology?

**Tip:** Understanding the concept of connectedness is crucial in topology, especially when dealing with properties like openness and closedness.

Prove the following propositions: 1. The set of interior points of a set A, denoted by int(A), is an open set. 2. The set of interior points is the largest open set contained in A. If G is open and int(A) ⊆ G ⊆ A, then int(A) = G. 3. The only subsets of ℝ that are both open and closed are ∅ and ℝ.

This page contains three key topology proofs: 1. The interior of a set is an open set. 2. The interior is the largest open set contained in a given set. 3. In ℝ, the only clopen sets are ∅ and ℝ. Explore the concepts of open and closed sets, interior points, and clopen sets in real analysis.

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