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.
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In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an -dimensional manifold, or -manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of -dimensional Euclidean space. One-dimensional manifolds include lines and circles, but not lemniscates. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane.
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.
In topology and related areas of mathematics, a subset A of a topological space X is said to be dense in X if every point of X either belongs to A or else is arbitrarily "close" to a member of A — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine approximation). Formally, is dense in if the smallest closed subset of containing is itself.
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