Mathematics Exam  >  Mathematics Videos  >  Mathematics for Competitive Exams  >  Open Set, Closed Set, Compact Set, Dense Set, Perfect Set, & Connected Set

Open Set, Closed Set, Compact Set, Dense Set, Perfect Set, & Connected Set Video Lecture | Mathematics for Competitive Exams

98 videos|27 docs|30 tests
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00:25 Open Set
09:20 Closed Set
17:40 Compact Set
19:02 Dense Set
22:10 Perfect Set
23:38 Connected Set
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FAQs on Open Set, Closed Set, Compact Set, Dense Set, Perfect Set, & Connected Set Video Lecture - Mathematics for Competitive Exams

1. What is the definition of an open set?
Ans. An open set is a subset of a topological space in which every point has a neighborhood contained entirely within the set. In other words, for every point in an open set, we can find a small ball around that point that is completely contained within the set.
2. How can a set be classified as closed?
Ans. A set is considered closed if it contains all its limit points. In other words, for every accumulation point of a set, the set itself must contain those points. A closed set can also be defined as the complement of an open set.
3. What does it mean for a set to be compact?
Ans. A set is said to be compact if it is both closed and bounded. This means that the set contains all its limit points and can be enclosed within a finite distance. In other words, every open cover of a compact set has a finite subcover.
4. What does it mean for a set to be dense?
Ans. A set is considered dense if its closure is equal to the entire space in which it is contained. In simpler terms, a set is dense if it is closely packed and fills up the space it occupies. Every point in the space can be arbitrarily close to a point in the dense set.
5. How is a perfect set defined?
Ans. A perfect set is a closed set that has no isolated points. This means that every point in a perfect set is a limit point of the set itself. In other words, there are no holes or gaps in the set, and every point is surrounded by other points in the set.
Video Timeline
Video Timeline
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00:25 Open Set
09:20 Closed Set
17:40 Compact Set
19:02 Dense Set
22:10 Perfect Set
23:38 Connected Set
More
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