Closure (topology)

In mathematics, 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 '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 mathematics, 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 '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. For S a subset of a Euclidean space, x is a point of closure of S if every open ball centered at x contains a point of S (this point may be x itself). This definition generalises to any subset S of a metric space X. Fully expressed, for X a metric space with metric d, x is a point of closure of S if for every r > 0, there is a y in S such that the distance d(x, y) < r. (Again, we may have x = y.) Another way to express this is to say that x is a point of closure of S if the distance d(x, S) := inf{d(x, s) : s in S} = 0. This definition generalises to topological spaces by replacing 'open ball' or 'ball' with 'neighbourhood'. Let S be a subset of a topological space X. Then x is a point of closure (or adherent point) of S if every neighbourhood of x contains a point of S. Note that this definition does not depend upon whether neighbourhoods are required to be open. The definition of a point of closure is closely related to the definition of a limit point. The difference between the two definitions is subtle but important — namely, in the definition of limit point, every neighbourhood of the point x in question must contain a point of the set other than x itself. Thus, every limit point is a point of closure, but not every point of closure is a limit point. A point of closure which is not a limit point is an isolated point. In other words, a point x is an isolated point of S if it is an element of S and if there is a neighbourhood of x which contains no other points of S other than x itself. For a given set S and point x, x is a point of closure of S if and only if x is an element of S or x is a limit point of S (or both). The closure of a set S is the set of all points of closure of S, that is, the set S together with all of its limit points. The closure of S is denoted cl(S), Cl(S), S ¯ {displaystyle scriptstyle {ar {S}}} or S − {displaystyle scriptstyle S^{-}} . The closure of a set has the following properties. Sometimes the second or third property above is taken as the definition of the topological closure, which still make sense when applied to other types of closures (see below).