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Neighbourhood of a Point and Deleted Neighbourhood of a Point Video Lecture | Mathematics for Competitive Exams
FAQs on Neighbourhood of a Point and Deleted Neighbourhood of a Point Video Lecture - Mathematics for Competitive Exams
| 1. What is the definition of a neighborhood of a point? |  |
Ans. In mathematics, a neighborhood of a point is a set that contains all the points near that point. It can be visualized as a small open interval or region surrounding the point. The neighborhood is often denoted as N(x), where x is the point.
| 2. How is a neighborhood of a point different from a deleted neighborhood of a point? | |
Ans. A neighborhood of a point includes the point itself, while a deleted neighborhood of a point excludes the point itself. In other words, a deleted neighborhood is obtained by removing the point from its corresponding neighborhood.
| 3. What does it mean for a point to be an interior point of a set? | |
Ans. A point is said to be an interior point of a set if it has a neighborhood completely contained within the set. In simpler terms, the point is surrounded by other points from the same set without venturing outside of it.
| 4. How are neighborhoods and deleted neighborhoods used in topology? | |
Ans. Neighborhoods and deleted neighborhoods are fundamental concepts in topology, which is the branch of mathematics that studies the properties of space. These concepts help define open sets, continuity, and convergence. They provide a framework for understanding the topological structure of spaces and the behavior of functions.
| 5. Can you provide an example of a neighborhood and a deleted neighborhood? | |
Ans. Sure! Let's consider the real number line. If we have a point x = 2, a neighborhood of this point could be the interval (1, 3), which includes the point itself. On the other hand, a deleted neighborhood of x = 2 could be the interval (1, 2) ∪ (2, 3), where the point 2 is excluded.
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