limit point

In mathematics, a limit point, also known as a point of accumulation, is a point related to a set of points. Specifically, every neighborhood around a limit point, no matter how small, contains another point that also belongs to the set. This concept is fundamental in topology, a branch of mathematics that generalizes Euclidean spaces by describing closeness in terms of set relationships rather than distance.12

A point p is considered a limit point of a set S if every open set containing p also contains at least one point from S, other than p itself. The idea of a limit is crucial for understanding continuity and derivatives in calculus. For instance, a function can approach a limit at a point where it is not defined, consistent with nearby values. A sequence of points can have at most one limit point if the topological space has the Hausdorff property, meaning any two points can be isolated in non-overlapping open sets.23