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Limit Point of a Set & Derived Set of a Set Video Lecture | Mathematics for Competitive Exams
FAQs on Limit Point of a Set & Derived Set of a Set Video Lecture - Mathematics for Competitive Exams
| 1. What is a limit point of a set? |  |
Ans. A limit point of a set is a point in a metric space such that every neighborhood of the point contains at least one point from the set other than the point itself. In other words, a limit point is a point that can be approximated arbitrarily closely by elements of the set.
| 2. What is the derived set of a set? | |
Ans. The derived set of a set is the set of all limit points of the given set. It includes all the points that can be approximated arbitrarily closely by elements of the set, including the set itself.
| 3. How is the derived set related to the limit points of a set? | |
Ans. The derived set and the set of limit points are essentially the same thing. They both refer to the set of points that can be approximated closely by elements of the given set. The derived set includes all the limit points, including the set itself.
| 4. Can a set have no limit points? | |
Ans. Yes, a set can have no limit points. In such cases, the derived set of the set will be empty. This occurs when the set is either finite or contains isolated points, i.e., points that have neighborhoods containing no other points from the set.
| 5. What is the importance of understanding limit points and derived sets in mathematics? | |
Ans. Understanding limit points and derived sets is crucial in various branches of mathematics, such as analysis and topology. These concepts help in defining continuity, compactness, and connectedness of subsets of metric spaces. They provide a foundation for studying convergence and continuity, which are fundamental in mathematical analysis. Additionally, limit points and derived sets play a significant role in the study of sequences, series, and topological properties of sets.
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