nonisolated

Nonisolated is a term used in topology to describe points that are not isolated within a given set. An isolated point of a set A in a topological space X is a point x in A for which there exists an open neighborhood U of x with U ∩ A = {x}. A nonisolated point is any point x in A for which every open neighborhood of x contains another point of A, so U ∩ (A \ {x}) is not empty for all neighborhoods U of x. In this sense, a nonisolated point is a kind of accumulation or limit point with respect to the set A.

The concept helps distinguish points that stand alone from those that cluster with other points of the set. A point can be nonisolated even if it is not a limit point of the entire space, as long as it is a limit point of the subset A within the ambient space. Equivalently, x is nonisolated in A if x ∈ A and every neighborhood of x contains some element of A other than x.

Examples illustrate the idea. In the real line with the standard topology, the set A = {1/n : n ∈ N} has every point 1/n isolated in A, while 0 is a limit point of A and is nonisolated with respect to the closure A ∪ {0}. In a discrete topology, where every singleton is open, all points are isolated, and there are no nonisolated points.

The notion is widely used in analysis and topology to study convergence, continuity, and density. It is common to refer to accumulation points as nonisolated points of a set, particularly when discussing properties that rely on the presence of nearby distinct points within the set.