Information AboutOpen Set |
| CATEGORIES ABOUT OPEN SET | |
| general topology | |
|
In other words, if x is surrounded only by elements of U; it can't be on the edge of U. As a typical example, consider the open Interval (0,1) consisting of all Real Number s x with 0 < x < 1. Here, the topology is the usual topology on the real line. If you "wiggle" such an x a little bit (but not too much), then the wiggled version will still be a number between 0 and 1. Therefore, the interval (0,1) is open. However, the interval Note that whether a given set U is open depends on the surrounding space, the "wiggle room". For instance, the set of Rational Number s between 0 and 1 (exclusive) is open ''in the rational numbers'', but it is not open ''in the real numbers''. Note also that "open" is not the opposite of " Closed " (a closed set is the Complement of an open set). First, there are sets which are both open and closed (called '' Clopen Set s''); in R and other Connected Space s, only the Empty Set and the whole space are clopen, while the set of all rational numbers smaller than √2 is clopen in the rationals. Also, there are sets which are neither open nor closed, such as DEFINITIONS The concept of open sets can be formalized in various degrees of generality. Function-analytic A point set in R''n'' is called ''open'' when every point ''P'' of the set is an Inner Point . Euclidean space A 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'') Intuitively, ε measures the size of the allowed "wiggles". An example of an open set in R2 (on a plane) would be all the points within a circle radius r, which satisfy the equation . Because the distance of any point p in this set from the edge of the set is greater than zero: , 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 spaces A 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 spaces In Topological Space s, 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.) USES Every 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. MANIFOLDS A 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 ALSO |
|
|