Check whether A' is a closed subset of X. Also check whether (A')'=A

### Problem Overview:
Let \( X \) be a topological space and \( A \) a subset of \( X \). The notation \( A' \) refers to the **derived set** of \( A \), which consists of all **limit points** of \( A \) in \( X \). The derived set \( A' \) may or may not be a closed subset of \( X \), and we are also interested in verifying whether \( (A')' = A \).

### Part 1: Checking if \( A' \) is a closed subset of \( X \)
A subset \( B \subseteq X \) is **closed** if and only if \( B = \overline{B} \), meaning \( B \) contains all its limit points (i.e., \( B' \subseteq B \)).

To check whether \( A' \) is closed, we need to verify if \( A' \) contains all its limit points:
- Take any limit point \( x \in (A')' \), which means every neighborhood of \( x \) contains points from \( A' \) other than \( x \) itself.
- If every limit point of \( A' \) is also contained within \( A' \), then \( A' \) is closed.

#### Conclusion for Part 1:
- In general, **\( A' \) is closed** in a topological space. This is because limit points of limit points are already included in \( A' \), so \( A' = \overline{A'} \).

### Part 2: Checking if \( (A')' = A \)
Next, we need to determine if the derived set of \( A' \), denoted \( (A')' \), is equal to the original set \( A \).

- **If \( A \) is finite**: The derived set \( A' \) is empty (since finite sets have no limit points), so \( (A')' = \emptyset \), which is clearly not equal to \( A \) unless \( A = \emptyset \).
  
- **If \( A \) is infinite**: Typically, \( A' \neq A \) unless \( A \) is already a closed set containing all its limit points (such as a perfect set). For example, if \( A \) is a discrete subset, its derived set might be empty, and \( (A')' \) will not equal \( A \).

#### Conclusion for Part 2:
- In general, \( (A')' \neq A \), unless \( A \) has some special properties (e.g., \( A \) is a closed set containing all its limit points, like a perfect set).

### Summary:
1. **\( A' \) is always a closed subset of \( X \).**
2. **\( (A')' = A \) does not hold in general**, but it can if \( A \) is already a perfect set (i.e., \( A = A' \)).

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Do you have any questions or need further clarification?

### 5 Related Questions:
1. What are examples of sets where \( (A')' = A \)?
2. Can you explain the concept of limit points in more detail?
3. How does the topology of \( X \) affect whether \( A' \) is closed?
4. What are perfect sets, and why do they satisfy \( A = A' \)?
5. How do we compute derived sets for specific examples like intervals or sequences?

### Tip:
Understanding the behavior of limit points and derived sets becomes easier when you explore different topologies (e.g., discrete, standard topology on \( \mathbb{R} \), etc.) and their effect on open and closed sets.

Check whether A' is a closed subset of X. Also check whether (A')'=A.

Explore how to determine if A' is a closed subset of X and whether (A')' equals A in topological spaces. This solution discusses limit points, derived sets, and closed sets, providing a detailed analysis suitable for undergraduate-level topology students.

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