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Encyclopedia > Open set

In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can "wiggle" or "change" any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U; it can't be on the edge of U. Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ... Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ...


As a typical 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. 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 (0,1] consisting of all numbers x with 0 < x ≤ 1 is not open; if you take x = 1 and move even the tiniest bit in the positive direction, you will be outside of (0,1]. In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... In mathematics, the set of real numbers, denoted R, or in blackboard bold , is the set of all rational and irrational numbers. ...


Note that whether a given set U is open depends on the surrounding space, the "wiggle room". For instance, the set of rational numbers 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 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. Also, there are sets which are neither open nor closed, such as (0,1] in R. In mathematics, a rational number (commonly called a fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In topology and related branches of mathematics, a closed set is a set whose complement is open. ... In set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement. ... In topology, a clopen set (or closed-open set) in a topological space is a set which is both open and closed. ... Connected and disconnected subspaces of R². The space A at top is connected; the shaded space B at bottom is not. ... In mathematics and more specifically set theory, the empty set is the unique set which contains no elements. ...

Example: The points satisfying are colored blue. The points satisfying are colored red. The red points form an open set. The union of the red and blue points is a closed set.
Example: The points satisfying r=sqrt{x^2+y^2} are colored blue. The points satisfying r<sqrt{x^2+y^2} are colored red. The red points form an open set. The union of the red and blue points is a closed set.

Contents

Image File history File links Red_blue_circle. ... Image File history File links Red_blue_circle. ...

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 inner point. In the theory of point sets, an inner point is any point which contains at least some neighbourhood of the set. ...


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) In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... In predicate logic, universal quantification is an attempt to formalise the notion that something (a logical predicate) is true for everything, or every relevant thing. ... In predicate logic, existential quantification is an attempt to formalize the notion that something (a logical predicate) is true for something, or at least one relevant thing. ... In mathematics, the Euclidean distance or Euclidean metric is the ordinary distance between the two points that one would measure with a ruler, which can be proven by repeated application of the Pythagorean theorem. ... In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. ...


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 of radius r, which satisfy the equation r>sqrt{x^2+y^2}.


Because the distance of any point p in this set from the edge of the set is greater than zero: r-sqrt{x^2+y^2}>0, 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) In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ...


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. 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. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ... 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, all elements of B that also belong to A), but no other elements. ... In mathematics the term countable set is used to describe the size of a set, e. ... In the mathematical field of topology a G-delta set or Gδ set is a set in a topological space which is in a certain sense simple. ...


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. A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. The relationship of one set being a subset of another is called inclusion. ... In mathematics, the interior of a set S consists of all points which are intuitively not on the edge of S. A point which is in the interior of S is an interior point of S. The notion of interior is in many ways dual to the notion of closure. ...


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. Partial plot of a function f. ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ... In mathematics, the image of an element x in a set X under the function f : X &#8594; Y, denoted by f(x), is the unique y in Y that is associated with x. ... In topology, an open map is a function between two topological spaces which maps open sets to open sets. ... Image of the Wikimedia Commons logo. ...


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 of real numbers. ...


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. On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ...


See also


  Results from FactBites:
 
Open set - Wikipedia, the free encyclopedia (895 words)
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 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.
(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).
Open - Wikipedia, the free encyclopedia (178 words)
Open (sport) - A type of competition in tennis and golf (among others) where entry is open to all qualifiers regardless of age.
In science, where an open system (system theory) means a system in which matter or energy can flow into and/or out of the system, in contrast to a closed system in which no energy or matter may enter or leave the system.
In computing, where an open system (computing) is a computer operating system that provides interoperability, portability or both, particularly Unix systems.
  More results at FactBites »

 
 

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