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Adherent Point & Closure of a Set Video Lecture | Mathematics for Competitive Exams
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FAQs on Adherent Point & Closure of a Set Video Lecture - Mathematics for Competitive Exams
| 1. What is the definition of adherent point? |  |
Ans. An adherent point of a set is a point that lies either in the set or in its closure. In other words, a point is an adherent point if every neighborhood of the point contains at least one point of the set.
| 2. How is closure of a set defined? | |
Ans. The closure of a set is the union of the set and all its limit points. It includes all the points that are either in the set or are adherent points of the set. The closure of a set can be denoted as the set itself with a bar on top, or by adding all the limit points to the set.
| 3. How are adherent points and closure related? | |
Ans. Adherent points and closure are closely related concepts. The closure of a set consists of all the adherent points of the set. In other words, the closure is the set itself along with any additional points that are adherent to the set.
| 4. Can a set have multiple closures? | |
Ans. No, a set can have only one closure. The closure of a set is a unique set that includes all the points that are either in the set or are adherent points. It is the smallest closed set that contains the original set.
| 5. Is the closure of a set always closed? | |
Ans. Yes, the closure of a set is always closed. By definition, the closure includes all the limit points of the set, which means that any sequence of points in the closure that converges will converge to a point in the closure itself. Therefore, the closure is closed.
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Dec 23, 2024 Last updated |
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