MalleusScientiarum said:As I recall an accumulation point is a point in the set A where any neighborhood of x has at least one other point in A that is not x.
An accumulation point is a point in a set S that can be approached arbitrarily closely by points in the set. This means that for any positive distance e, there exists a point in the set within e units of the accumulation point.
While accumulation points and limit points are often used interchangeably, there is a subtle difference between the two. An accumulation point is a point in a set that can be approached by other points in the set, while a limit point is a point in a set where every neighborhood of the point contains infinitely many points from the set.
The distance e must be greater than 0 in order to ensure that there is at least one point in the set S within e units of the accumulation point. If e was equal to 0, then there would be no guarantee that there are any points in the set within a certain distance of the accumulation point.
No, an accumulation point cannot be an isolated point. An isolated point is a point in a set that has a neighborhood that contains no other points from the set. However, an accumulation point, by definition, must have points from the set arbitrarily close to it, meaning it cannot be isolated.
Accumulation points are important in mathematical analysis as they help define the concepts of convergence, continuity, and compactness. They also play a crucial role in the proof of the Bolzano-Weierstrass theorem, which states that every bounded sequence has a convergent subsequence.