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. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In other words, the distance between any point x in U and the edge of U is always greater than zero. Distance is a numerical description of how far apart objects are

As an example, consider the open interval (0,1) consisting of all real numbers x with 0 < x < 1. In Mathematics, an interval is a set of Real numbers with the property that any number that lies between two numbers in the set is also included in the set In Mathematics, the real numbers may be described informally in several different ways 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].

Example: The points (x,y) satisfying x² + y² = r² are colored blue. The points (x,y) satisfying x² + y² < r² are colored red. The red points form an open set. The union of the red and blue points is a closed set.

Definitions

The concept of open sets can be formalized in various degrees of generality.

Function-analytic

A point set in Rⁿ is called open when every point P of the set is an interior point. In Mathematics, the interior of a set S consists of all points of S that are intuitively "not on the edge of S "

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. In Predicate logic, universal quantification is an attempt to formalize the notion that something (a Logical predicate) is true for everything, or every In Predicate logic, an existential quantification is the predication of a property or relation to at least one member of the domain In Mathematics, the Euclidean distance or Euclidean metric is the "ordinary" Distance between two points that one would measure with a ruler Equivalently, U is open if every point in U has a neighbourhood contained in U. In Topology and related areas of Mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a Topological space.

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. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined (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 spaces, the concept of openness is taken to be fundamental. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. 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. 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 G_δ sets. In the Mathematical field of Topology, a Gδ set, is a Subset of a Topological space that is a countable intersection of open 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. )

Properties

• The empty set is open. In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members
• The union of any number of open sets is open. In Set theory, the term Union (denoted as ∪ refers to a set operation used in the convergence of set elements to form a resultant set containing the elements of both sets
• The intersection of a finite number of open sets is open. In Mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently

Uses

Open sets have a fundamental importance in the branch of topology. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of 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. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined In the Mathematical field of Topology, a uniform space is a set with a uniform structure.

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. In Mathematics, the interior of a set S consists of all points of S that are intuitively "not on the edge of S " 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 Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage The map f is called open if the image of every open set in X is open in Y. In Topology, an open map is a function between two Topological spaces which maps Open sets to open sets In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage

An open set on the real line has the characteristic property that it is a countable union of disjoint open intervals. In Mathematics, the real line is simply the set R of singleton Real numbers However this term is usually used when R is to be treated as a

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. In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions In Mathematics, the real numbers may be described informally in several different ways 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. In Topology, a clopen set (or closed-open set, a Portmanteau word in a Topological space is a set which is both open and closed In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members 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. In Set theory, the term Union (denoted as ∪ refers to a set operation used in the convergence of set elements to form a resultant set containing the elements of both sets 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. In Topology and related branches of Mathematics, a closed set is a set whose complement is open. In Discrete mathematics and predominantly in Set theory, a complement is a concept used in comparisons of sets to refer to the unique values of one set in relation

See also

• Closed set
• Clopen set
• Neighbourhood

External links

• Open Set on PlanetMath