A limit point, also known as an accumulation point, is a point in a set where every open neighborhood of that point contains infinitely many points of the set. In other words, if we draw a small circle around the limit point, it will always contain points from the set.
In real analysis, limit points and cluster points are essentially the same. Both refer to points in a set that are surrounded by infinitely many points of the set. However, the term "limit point" is more commonly used in real analysis, whereas "cluster point" is used more in general topology.
No, a finite set cannot have limit points. This is because a limit point requires an infinite number of points around it, and a finite set, by definition, has a finite number of points.
Limit points play a crucial role in defining concepts such as continuity, compactness, and connectedness in real analysis. They also help in understanding the behavior of sequences and series in real analysis.
No, a set can have multiple limit points. For example, in the set of real numbers, any point on the real number line can be considered a limit point. However, some sets may have no limit points, such as the set of rational numbers in the interval [0,1].