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Encyclopedia > Closed set

In topology and related branches of mathematics, a closed set is a set whose complement is open. A Möbius strip, a surface with only one side and one edge; such shapes are an object of study in topology. ... 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. ... In set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement. ... 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...

Contents

Definition of a closed set

A set is closed if every limit point of the set is a point in the set. In mathematics, informally speaking, a limit point (or cluster point) of a set S in a topological space X is a point x in X that can be approximated by points of S other than x as well as one pleases. ...


Properties of closed sets

A closed set contains its own boundary. In other words, if you are "outside" a closed set and you "wiggle" a little bit, you will stay outside the set. In topology, the boundary of a subset S of a topological space X is the sets closure minus its interior. ...


Any intersection of arbitrarily many closed sets is closed, and any union of finitely many closed sets is closed. In particular, the empty set and the whole space are closed. In fact, given a set X and a collection F of subsets of X that has these properties, then F will be the collection of closed sets for a unique topology on X. The intersection property also allows one to define the closure of a set A in a space X, which is defined as the smallest closed subset of X that is a superset of A. Specifically, the closure of A can be constructed as the intersection of all of these closed supersets. 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 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, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... In mathematics and more specifically set theory, the empty set is the unique set which contains no elements. ... In mathematics, the closure of a set S consists of all points which are intuitively close to S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior. ... 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. ...


Sets that can be constructed as the union of countably many closed sets are denoted Fσ sets. These sets need not be closed. In mathematics the term countable set is used to describe the size of a set, e. ... In mathematics, an Fσ set (said F-sigma set) is a countable union of closed sets. ...


Examples of closed sets

  • The closed interval [a,b] of real numbers is closed. (see the entry on intervals for an explanation of the bracket and parenthesis set notation.)
  • The unit interval [0,1] is closed in the real numbers, and the set [0,1] ∩ Q of rational numbers between 0 and 1 (inclusive) is closed in the space of rational numbers, but [0,1] ∩ Q is not closed in the real numbers.
  • Some sets are neither open nor closed, for instance the half-open interval [0,1) in the real numbers.
  • Some sets are both open and closed and are called clopen sets.

The term interval is used in the following contexts: cricket mathematics music time This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... In mathematics, the real numbers may be described informally in several different ways. ... The term interval is used in the following contexts: cricket mathematics music time This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... In mathematics, the unit interval is the interval [0,1], that is the set of all real numbers x such that zero is less than or equal to x and x is less than or equal to one. ... In mathematics, the real numbers may be described informally in several different ways. ... 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 mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... 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. ...

More about closed sets

In functional analysis, a point set is closed if it contains all its boundary points. In topology, the boundary of a subset S of a topological space X is the sets closure minus its interior. ...


The notion of closed set is defined above in terms of open sets, a concept that makes sense for topological spaces, as well as for other spaces that carry topological structures, such as metric spaces, differentiable manifolds, uniform spaces, and gauge spaces. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... This article or section is in need of attention from an expert on the subject. ... In the mathematical field of topology, a uniform space is a set with a uniform structure. ... In topology and related areas of mathematics a gauge space is a topological space where the topology is defined by a family of pseudometrics. ...


An alternative characterization of closed sets is available via sequences and nets. A subset A of a topological space X is closed in X if and only if every limit of every net of elements of A also belongs to A. In a first-countable space (such as a metric space), it is enough to consider only sequences, instead of all nets. One value of this characterisation is that it may be used as a definition in the context of convergence spaces, which are more general than topological spaces. Notice that this characterisation also depends on the surrounding space X, because whether or not a sequence or net converges in X depends on what points are present in X. In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ... In topology and related areas of mathematics a net or Moore-Smith sequence is a generalization of a sequence, intended to unify the various notions of limit and generalize them to arbitrary topological spaces. ... In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as their index increases indefinitely. ... In topology, a first-countable space is a topological space satisfying the first axiom of countability. Specifically, a space X is said to be first-countable if each point has a countable local base. ... In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ...


We have seen twice that whether a set is closed is relative; it depends on the space that it's embedded in. However, the compact Hausdorff spaces are "absolutely closed" in a certain sense.To be precise, if you embed a compact Hausdorff space K in an arbitrary Hausdorff space X, then K will always be a closed subset of X; the "surrounding space" does not matter here. In fact, this property characterizes the compact Hausdorff spaces. Stone-Čech compactification, a process that turns a completely regular Hausdorff space into a compact Hausdorff space, may be described as adjoining limits of certain nonconvergent nets to the space. In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... In mathematics, compactification is applied to topological spaces to make them compact spaces. ... In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are particularly nice kinds of topological spaces. ...


See also


  Results from FactBites:
 
Closed set - Wikipedia, the free encyclopedia (572 words)
In topology and related branches of mathematics, a closed set is a set whose complement is open.
The notion of closed set is defined above in terms of open sets, a concept that makes sense for topological spaces, as well as for other spaces that carry topological structures, such as metric spaces, differentiable manifolds, uniform spaces, and gauge spaces.
We have seen twice that whether a set is closed is relative; it depends on the space that it's embedded in.
Encyclopedia: Closed set (1544 words)
In mathematics, the closure of a set S consists of all points which are intuitively close to S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior.
In mathematics the term countable set is used to describe the size of a set, e.
In mathematics, an Fσ set (said F-sigma set) is a countable union of closed sets.
  More results at FactBites »

 
 

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