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Image:Red blue circle.png Example: The points satisfying <math>r=\sqrt{x^2+y^2}</math> are colored blue. The points satisfying <math>r<\sqrt{x^2+y^2}</math> are colored red. The red points form an open set. The union of the red and blue points is a closed set.
DefinitionsThe concept of open sets can be formalized in various degrees of generality. Function-analyticA point set in Rn is called open when every point P of the set is an inner point. Euclidean spaceA subset U of the Euclidean n-space Rn is called open if, given any point x in U, there exists a real number ε > 0 such that, given any point y in Rn whose Euclidean distance from x is smaller than ε, y also belongs to U. (Equivalently, U is open if every point in U has a neighbourhood contained in U)
An example of an open set in R2 (on a plane) would be all the points within a circle of radius r, which satisfy the inequality <math>r>\sqrt{x^2+y^2}</math>. Because the distance of any point p in this set from the edge of the set is greater than zero: <math>r-\sqrt{x^2+y^2}>0</math>, we can set ε to half of this distance, which means ε is also greater than zero, and all the points that are within a distance of ε to p are also in the set, thus satisfying the conditions for an open set. Metric spacesA subset U of a metric space (M,d) is called open if, given any point x in U, there exists a real number ε > 0 such that, given any point y in M with d(x,y) < ε, y also belongs to U. (Equivalently, U is open if every point in U has a neighbourhood contained in U) This generalises the Euclidean space example, since Euclidean space with the Euclidean distance is a metric space. Topological spacesIn topological spaces, the concept of openness is taken to be fundamental. One starts with an arbitrary set X and a family of subsets of X satisfying certain properties that every "reasonable" notion of openness is supposed to have. (Specifically: the union of open sets is open, the finite intersection of open sets is open, and in particular the empty set and X itself are open.) Such a family T of subsets is called a topology on X, and the members of the family are called the open sets of the topological space (X,T). Note that infinite intersections of open sets need not be open. Sets that can be constructed as the intersection of countably many open sets are denoted Gδ sets. The topological definition of open sets generalises the metric space definition: If you start with a metric space and define open sets as before, then the family of all open sets will form a topology on the metric space. Every metric space is hence in a natural way a topological space. (There are however topological spaces which are not metric spaces.) UsesEvery subset A of a topological space X contains a (possibly empty) open set; the largest such open set is called the interior of A. It can be constructed by taking the union of all the open sets contained in A. Given topological spaces X and Y, a function f from X to Y is continuous if the preimage of every open set in Y is open in X. The map f is called open if the image of every open set in X is open in Y. An open set on the real line has the characteristic property that it is a countable union of disjoint open intervals. ManifoldsA manifold is called open if it is a manifold without boundary and if it is not compact. This notion differs somewhat from the openness discussed above. See alsocs:Otevřená množina de:Offene Menge es:Conjunto abierto eo:Malfermita aro fr:Ouvert (topologie) ko:열린 집합 is:Opið mengi it:Insieme aperto he:קבוצה פתוחה nl:Open verzameling ja:開集合 pl:Zbiór otwarty pt:Conjunto aberto ru:Открытое множество sk:Otvorená množina sv:Öppen mängd zh:开集 zh-classical:開集
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