Wednesday, January 9, 2008
In topology and related fields of mathematics, a set U is called open if, intuitively speaking, starting from any point x in U one can move by a small amount in any direction and still be in the set U. In other words, the distance between any point x in U and the edge of U is always greater than zero.
As an example, consider the open interval (0,1) consisting of all real numbers x with 0 < x < 1. Here, the topology is the usual topology on the real line. We can look at this in two ways. Since any point in the interval is different from 0 and 1, the distance from that point to the edge is always non-zero. Or equivalently, for any point in the interval we can move by a small enough amount in any direction without touching the edge and still be inside the set. Therefore, the interval (0,1) is open. However, the interval (0,1] consisting of all numbers x with 0 < x ≤ 1 is not open; if one takes x = 1 and moves even the tiniest bit in the positive direction, one will be outside of (0,1].
A point set in R is called open when every point P of the set is an inner point.
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.
In 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. 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.)
The empty set is open.
The union of countably many open sets is open.
The intersection of a finite set of open sets is open. Properties
Open sets have a fundamental importance in the branch of topology. The concept is required to define and make sense for topological space and other topological structures that deal with the notions of closeness and convergence for a space such as metric spaces and uniform spaces.
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.
Posted by bushganizer258 at 8:30 AM