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
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.
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