The **closure** of a set \( S \) in a topological space is the smallest closed set that contains \( S \). It can be thought of as the set \( S \) together with all its **limit points** (points that can be approached by sequences from \( S \)). The closure of a set is often denoted by \( \overline{S} \) or \( \text{cl}(S) \).

### Definition in Different Contexts:
1. **In Euclidean Space**: If \( S \) is a subset of the real numbers \( \mathbb{R} \), the closure of \( S \) is \( S \) plus any limit points of \( S \). For instance, if \( S = (0, 1) \), then the closure of \( S \) is \( [0, 1] \).

2. **In Metric Spaces**: The closure of \( S \) includes all points that are limits of sequences in \( S \), even if those points are not originally in \( S \).

3. **In General Topology**: The closure of \( S \) is the intersection of all closed sets containing \( S \).

### Key Properties:
- The closure of \( S \) is a closed set.
- \( S \subseteq \overline{S} \).
- If \( S \) is already closed, then \( \overline{S} = S \).
- The closure can be constructed by adding all limit points of \( S \) to the set \( S \).

### Example:
If \( S = \{x \in \mathbb{R} : 0 < x < 1\} \), then the closure of \( S \) is \( \overline{S} = [0, 1] \) because 0 and 1 are limit points of \( S \).

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### Related Questions:
1. What are limit points and how are they related to the closure of a set?
2. How is the closure of a set defined in different topological spaces?
3. What is the difference between the closure of a set and its interior?
4. How does the closure relate to open sets?
5. Can the closure of a set be empty?
6. What are some examples of closure in different types of spaces, like discrete or metric spaces?
7. How is the closure operation used in analysis and topology?
8. What is the closure of a set of rational numbers in \( \mathbb{R} \)?

#### Tip:
To find the closure of a set, always consider the limit points and whether the set already contains its boundaries.

Explore the definition and properties of the closure of a set in topological spaces. Learn how the closure of a set is defined as the smallest closed set containing the set itself, including all limit points. Examples and key properties are covered.

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