In topology and related branches of mathematics, a closed set is a set whose complement is open. 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 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 In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in
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Definition of a closed set
In a topological space, a set is closed if and only if it coincides with its closure. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Mathematics, the closure of a set S consists of all points which are intuitively "close to S " Equivalently, a set is closed if and only if it contains all of its limit points. In Mathematics, informally speaking a limit point of a set S in a Topological space X is a point x in X that can be "approximated"
This is not to be confused with a closed manifold. See also Classification of manifolds#Point-set In Mathematics, a closed manifold is type of Topological space, namely a compact
Properties of closed sets
A closed set contains its own boundary. For a different notion of boundary related to Manifolds see that article In other words, if you are "outside" a closed set and you "wiggle" a little bit, you will stay outside the set. Note that this is also true if the boundary is the empty set, e. g. in the metric space of rational numbers, for the set of numbers of which the square is less than 2.
Any intersection of arbitrarily many closed sets is closed, and any union of finitely many closed sets is closed. 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 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 In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2. In particular, the empty set and the whole space are closed. In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members 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. In Mathematics, the closure of a set S consists of all points which are intuitively "close to S " Specifically, the closure of A can be constructed as the intersection of all of these closed supersets.
Sets that can be constructed as the union of countably many closed sets are denoted Fσ sets. In mathematics an Fσ set (said F-sigma set) is a Countable union of Closed sets The notation originated in France with These sets need not be closed.
Examples of closed sets
- The closed interval [a,b] of real numbers is closed. In Mathematics, the real numbers may be described informally in several different ways (See intervals for an explanation of the bracket and parenthesis set notation. )
- The unit interval [0,1] is closed in the metric space 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. In Mathematics, the unit interval is the interval, that is the set of all Real numbers x such that zero is less than or equal to x 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
- Some sets are neither open nor closed, for instance the half-open interval [0,1) in the real numbers. 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
- Some sets are both open and closed and are called clopen sets. 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 point set topology a set A is closed if it contains all its boundary points. In Mathematics, general topology or point-set topology is the branch of Topology which studies properties of Topological spaces and structures For a different notion of boundary related to Manifolds see that article
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 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 A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. 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
An alternative characterization of closed sets is available via sequences and nets. In Mathematics, a sequence is an ordered list of objects (or events This article is about nets in Topological spaces and not about ε-nets in Metric spaces In Topology and related areas of Mathematics 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 Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" In a first-countable space (such as a metric space), it is enough to consider only sequences, instead of all nets. In Topology, a branch of Mathematics, a first-countable space is a Topological space satisfying the "first Axiom of countability " In Mathematics, a sequence is an ordered list of objects (or events 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.
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. In Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological space 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 the mathematical discipline of General topology, Stone–Čech compactification is a technique for constructing a universal map from a topological space X to a In Topology and related branches of Mathematics, Tychonoff spaces and completely regular spaces are particularly nice kinds of Topological spaces
See also
In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x inDapyx Software network: MP3 Explorer | Ebook Manager | Zenithic