let X be a metric space. show that every subset of X is open if and only if each subset of X which consists of a single point is open

To prove that every subset of a metric space \( X \) is open if and only if each subset of \( X \) that consists of a single point is open, let's break down the proof into two directions:

### Direction 1: If every subset of \( X \) is open, then each subset of \( X \) consisting of a single point is open.

1. **Assumption**: Every subset of \( X \) is open.
   
2. Consider a subset \( \{x\} \) where \( x \in X \). By the assumption, \( \{x\} \) is a subset of \( X \), and since every subset of \( X \) is open, \( \{x\} \) must be open.

3. **Conclusion**: Therefore, each subset of \( X \) that consists of a single point is open.

### Direction 2: If each subset of \( X \) that consists of a single point is open, then every subset of \( X \) is open.

1. **Assumption**: Each subset of \( X \) consisting of a single point is open.
   
2. We want to show that every subset of \( X \) is open. Consider any subset \( A \subseteq X \). To prove that \( A \) is open, take any point \( x \in A \). By the assumption, \( \{x\} \) is open.

3. In a metric space, if \( \{x\} \) is open, then for each \( x \in A \), there exists an open ball \( B(x, r) \subseteq \{x\} \) for some radius \( r > 0 \).

4. Since \( \{x\} \) is open for each \( x \in A \), every point in \( A \) is an interior point. Thus, \( A \) is open.

5. Since \( A \) was arbitrary, this implies that every subset of \( X \) is open.

### Conclusion
- We have shown that if every subset of \( X \) is open, then each subset consisting of a single point is open.
- Conversely, if each subset consisting of a single point is open, then every subset of \( X \) is open.

Thus, the two conditions are equivalent.

Would you like more details or have any questions? Here are five questions to deepen your understanding:

1. What is the definition of an open set in a metric space?
2. Can you explain why an open set must contain an open ball around each of its points?
3. How does the concept of open sets relate to the topology of a metric space?
4. What are the implications if every subset of a metric space is open for the structure of that space?
5. Can you give an example of a metric space where only single-point sets are open?

**Tip:** In topology, understanding the properties of open and closed sets helps in grasping the underlying structure of different spaces.

Explore the proof that every subset of a metric space is open if and only if each subset consisting of a single point is open. Understand the fundamental concepts of metric spaces, topology, and open sets with detailed step-by-step reasoning.

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