Question
What does it mean for a set to be “closed”?
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GPT-4o mini
In mathematics, particularly in the context of topology, a set is considered "closed" if it contains all its limit points. This means that if a sequence of points within the set converges to a limit, that limit must also be included within the set.
In more practical terms:
- **In Euclidean space**: A subset \(C\) of \(\mathbb{R}^n\) is closed if it contains all points that can be approached by points of \(C\). For example, the closed interval \([a, b]\) in \(\mathbb{R}\) is closed because it includes the endpoints \(a\) and \(b\).
- **Complement**: Another way to define closed sets is through their complement. A set \(C\) is closed if its complement \( \mathbb{R}^n \setminus C \) is open.
- **Examples**:
- The set of real numbers, \(\mathbb{R}\), is closed.
- The set of all continuous functions over a closed interval is also closed.
- **Properties**: Closed sets have several interesting properties:
- The intersection of any collection of closed sets is closed.
- The finite union of closed sets is also closed.
In summary, a closed set is one that includes all points that can be "approached" from within the set, ensuring that the limit of sequences or functions from the set remains within it.
In more practical terms:
- **In Euclidean space**: A subset \(C\) of \(\mathbb{R}^n\) is closed if it contains all points that can be approached by points of \(C\). For example, the closed interval \([a, b]\) in \(\mathbb{R}\) is closed because it includes the endpoints \(a\) and \(b\).
- **Complement**: Another way to define closed sets is through their complement. A set \(C\) is closed if its complement \( \mathbb{R}^n \setminus C \) is open.
- **Examples**:
- The set of real numbers, \(\mathbb{R}\), is closed.
- The set of all continuous functions over a closed interval is also closed.
- **Properties**: Closed sets have several interesting properties:
- The intersection of any collection of closed sets is closed.
- The finite union of closed sets is also closed.
In summary, a closed set is one that includes all points that can be "approached" from within the set, ensuring that the limit of sequences or functions from the set remains within it.
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