Closure (topology)

In topology, the closure of a subset S of points in a topological space consists of all points in S together with all limit points of S. The closure of S may equivalently be defined as the union of S and its boundary, and also as the intersection of all closed sets containing S. Intuitively, the closure can be thought of as all the points that are either in S or "very near" S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior. Adherent point For as a subset of a Euclidean space, is a point of closure of if every open ball centered at contains a point of (this point can be itself). This definition generalizes to any subset of a metric space Fully expressed, for as a metric space with metric is a point of closure of if for every there exists some such that the distance ( is allowed). Another way to express this is to say that is a point of closure of if the distance where is the infimum. This definition generalizes to topological spaces by replacing "open ball" or "ball" with "neighbourhood". Let be a subset of a topological space Then is a or of if every neighbourhood of contains a point of (again, for is allowed). Note that this definition does not depend upon whether neighbourhoods are required to be open. Limit point of a set The definition of a point of closure of a set is closely related to the definition of a limit point of a set. The difference between the two definitions is subtle but important – namely, in the definition of a limit point of a set , every neighbourhood of must contain a point of , i.e., each neighbourhood of obviously has but it also must have a point of that is not equal to in order for to be a limit point of . A limit point of has more strict condition than a point of closure of in the definitions. The set of all limit points of a set is called the . A limit point of a set is also called cluster point or accumulation point of the set. Thus, every limit point is a point of closure, but not every point of closure is a limit point.