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:
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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 \(b\).
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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.
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Examples:
- The set of real numbers, \(\mathbb{R}\), is closed.
- The set of all continuous functions over a closed interval is also closed.
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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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