Open set
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In mathematics, more specifically point-set topology and metric topology, the notion of an open set provides a fundamental way to speak of distance in a topological space, without explicitly defining a metric on the space. In particular, although one cannot obtain concrete values for the distance between two points in a topological space, one may still be able to speak of "nearness" in the space, thus allowing concepts such as continuity to translate into the theory of open sets.
Intuitively (see below for a more intuitive discussion), an open set provides one a method to distinguish two points. For example, if about one point in a topological space, there exists an open set not containing another (distinct) point, the two points are referred to as topologically distinguishable. In this manner, one may speak of whether two subsets of a topological space are "near" without concretely defining a metric on the topological space. Therefore, topological spaces may be seen as a generalization of metric spaces.
Point-set topology is the area of mathematics concerned with general topological spaces, and the relations between them. In the category of topological spaces, morphisms are continuous functions between topological spaces. Continuous functions are readily observed to preserve topological structure, as they map "points close together" to "points close together"; that is, they preserve the structure of open sets defined on the space.
In metric topology, one can concretely define a distance function between two points, and thus metric spaces also have a topology, i.e. a certain structure of open sets defined on them. Thus as opposed to the pure topological invariants, metric topology deals with isometries and the like; that is, distance preserving maps. In this case, the idea of an open set is used as an organizational tool rather than an object of study. From the topological point of view, metric spaces are fairly well understood, although many open problems still remain in metrizability theory.
The concept of an open set is of fundamental importance in mathematics due to the numerous areas which exploit it. In algebraic geometry, the Zariski topology is a certain family of open sets which in some sense, reflect the algebraic nature of varieties. In differential topology, the open sets of a topological space are often required to have a simple structure - that is, each point within the space is required to have a neighbourhood homeomorphic to an open ball in finite-dimensional Euclidean space. Although open sets are of central importance (and are studied) in point-set topology, they are also used as an organizational tool in other important branches of mathematics.
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[edit] Informal discussion
A subset U is open if the distance between any point x in U and the edge of U is greater than zero: starting from any point x in U one can move by a small amount in any direction and still be in the set U.
As an example, consider a set defined by the interval (0, 1) consisting of all real numbers x with 0 < x < 1. Here, the topology is the usual topology on the real line. Now, the distance between any point in the interval (i.e. any member of the set) and the "edges" by which the interval is defined (i.e. 0 and 1) cannot be zero, since neither 0 nor 1 are contained in the set. 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 in the real line.
As a counterexample, consider the interval (0, 1] consisting of all numbers x with 0 < x ≤ 1. It is not an open subset of the real line; if one takes x = 1 and moves any amount in the positive direction, one will be outside of (0, 1].
[edit] Definitions
The concept of open sets can be formalized with various degrees of generality, for example:
[edit] Geometric
A point set in Rn is called open when every point P of the set is an interior point.
[edit] 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.
[edit] 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.
[edit] Topological spaces
If a nonempty set X has a collection of subsets T that is a topological space, then any member of T is an open set.
Note that infinite intersections of open sets need not be open. For example, the intersection of all intervals of the form (−1/n, 1/n), where n is a positive integer, is the set {0} which is closed in the real line. 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 one begins with a metric space and defines open sets as before, then the family of all open sets is a topology on the metric space. Every metric space is therefore, in a natural way, a topological space. There are, however, topological spaces that are not metric spaces.
[edit] Properties
- The empty set is both open and closed.
- The union of any number of open sets is open.
- The intersection of a finite number of open sets is open.
[edit] Uses
Open sets have a fundamental importance in topology. The concept is required to define and make sense of topological space and other topological structures that deal with the notions of closeness and convergence for spaces 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.
[edit] Note
Note that whether a given set U is open depends on the surrounding space. For instance, if U is defined as the set of rational numbers in the interval (0, 1), then U is open in the rational numbers, but not open in the real numbers. This is because when U is in the rational numbers there are no irrational numbers that can be moved to—the smallest possible displacement is from one rational number to another. Also, no matter how close an element of U is to 0 or 1, there is always another rational number between it and 0 or 1, so from any element of U there is always a way to make a small enough displacement that you can get closer to 0 or 1 while staying inside U. But, when this set is in the real numbers, there are irrational numbers between all of the rational numbers and it is possible to move from an element of U to an irrational number (which is not an element of U). So, for any displacement from some beginning element of U to some ending element, there is always a smaller distance from the beginning element to an irrational number which is outside of U. (Even though the irrational number may be between 0 and 1, it is not in U because U contains only rational numbers.)
Some sets are both open and closed (called clopen sets); in R and other connected spaces, 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. While others are neither open nor closed, such as (0, 1] in R. In fact, the set (0, 1] is the union of the sets (0, 1) and {1}, an open set and a closed set respectively. An important point is that an open set is not the opposite of "closed set", rather a closed set is the complement of an open set.
