limitpoint

In mathematics, a limit point, also known as an accumulation point or cluster point, of a set is a point that can be "approached" arbitrarily closely by other points in the set. More formally, a point p is a limit point of a set S if every neighborhood of p contains at least one point of S other than p itself. A neighborhood of a point p is typically defined as an open interval or open ball centered at p, with a radius greater than zero.

This concept is fundamental in topology and analysis. A set can be finite or infinite, and its limit points capture the notion of points where the set "accumulates." For example, in the set of real numbers, the limit points of the open interval (0, 1) are all the points in the closed interval [0, 1]. The endpoints 0 and 1 are limit points because any neighborhood around them will contain points from within the interval (0, 1).

The set of all limit points of a set S is called its derived set, often denoted by S'. A set is considered closed if it contains all of its limit points. Conversely, a set is open if none of its points are limit points of its complement. The concept of limit points is crucial for understanding continuity, convergence, and the structure of various mathematical spaces.