In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set.In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a closed set is a set which is closed under the limit operation. This should not be confused with a closed manifold.
By definition, a subset of a topological space is called closed if its complement is an open subset of ; that is, if A set is closed in if and only if it is equal to its closure in Equivalently, a set is closed if and only if it contains all of its limit points. Yet another equivalent definition is that a set is closed if and only if it contains all of its boundary points. Every subset is always contained in its (topological) closure in which is denoted by that is, if then Moreover, is a closed subset of if and only if
An alternative characterization of closed sets is available via sequences and nets. A subset of a topological space is closed in if and only if every limit of every net of elements of also belongs to In a first-countable space (such as a metric space), it is enough to consider only convergent sequences, instead of all nets. One value of this characterization 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 characterization also depends on the surrounding space because whether or not a sequence or net converges in depends on what points are present in A point in is said to be close to a subset if (or equivalently, if belongs to the closure of in the topological subspace meaning where is endowed with the subspace topology induced on it by ). Because the closure of in is thus the set of all points in that are close to this terminology allows for a plain English description of closed subsets:
In terms of net convergence, a point is close to a subset if and only if there exists some net (valued) in that converges to If is a topological subspace of some other topological space in which case is called a topological super-space of then there might exist some point in that is close to (although not an element of ), which is how it is possible for a subset to be closed in but to not be closed in the "larger" surrounding super-space If and if is any topological super-space of then is always a (potentially proper) subset of which denotes the closure of in indeed, even if is a closed subset of (which happens if and only if ), it is nevertheless still possible for to be a proper subset of However, is a closed subset of if and only if for some (or equivalently, for every) topological super-space of
Closed sets can also be used to characterize continuous functions: a map is continuous if and only if for every subset ; this can be reworded in plain English as: is continuous if and only if for every subset maps points that are close to to points that are close to Similarly, is continuous at a fixed given point if and only if whenever is close to a subset then is close to
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.
Whether a set is closed depends on the space in which it is embedded. However, the compact Hausdorff spaces are "absolutely closed", in the sense that, if you embed a compact Hausdorff space in an arbitrary Hausdorff space then will always be a closed subset of ; the "surrounding space" does not matter here. 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.
Furthermore, every closed subset of a compact space is compact, and every compact subspace of a Hausdorff space is closed.
Closed sets also give a useful characterization of compactness: a topological space is compact if and only if every collection of nonempty closed subsets of with empty intersection admits a finite subcollection with empty intersection.
A topological space is disconnected if there exist disjoint, nonempty, open subsets and of whose union is Furthermore, is totally disconnected if it has an open basis consisting of closed sets.
A closed set contains its own boundary. In other words, if you are "outside" a closed set, you may move a small amount in any direction and still 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
In fact, if given a set and a collection of subsets of such that the elements of have the properties listed above, then there exists a unique topology on such that the closed subsets of are exactly those sets that belong to The intersection property also allows one to define the closure of a set in a space which is defined as the smallest closed subset of that is a superset of Specifically, the closure of 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. These sets need not be closed.
In mathematics, a continuous function is a function that does not have any abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its output can be assured by restricting to sufficiently small changes in its input. If not continuous, a function is said to be discontinuous. Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, during which attempts such as the epsilon–delta definition were made to formalize it.
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more obvious, topology called the box topology, which can also be given to a product space and which agrees with the product topology when the product is over only finitely many spaces. However, the product topology is "correct" in that it makes the product space a categorical product of its factors, whereas the box topology is too fine; in that sense the product topology is the natural topology on the Cartesian product.
This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric topology.
In mathematics, open sets are a generalization of open intervals in the real line. In a metric space—that is, when a distance is defined—open sets are the sets that, with every point P, contain all points that are sufficiently near to P.
In mathematics, the closure of a subset S of points in a topological space consists of all points in S together with all limit points of S. The closure of S may equivalently be defined as the union of S and its boundary, and also as the intersection of all closed sets containing S. Intuitively, the closure can be thought of as all the points that are either in S or "near" 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, a topological vector space is one of the basic structures investigated in functional analysis. A topological vector space is a vector space which is also a topological space, this implies that vector space operations be continuous functions. More specifically, its topological space has a uniform topological structure, allowing a notion of uniform convergence.
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology.
In mathematics, a Baire space is a topological space such that every intersection of a countable collection of open dense sets in the space is also dense. Complete metric spaces and locally compact Hausdorff spaces are examples of Baire spaces according to the Baire category theorem. The spaces are named in honor of René-Louis Baire who introduced the concept.
In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. That is, a function is open if for any open set in the image is open in Likewise, a closed map is a function that maps closed sets to closed sets. A map may be open, closed, both, or neither; in particular, an open map need not be closed and vice versa.
In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a topology induced from that of X called the subspace topology.
In general topology and related areas of mathematics, the final topology on a set with respect to a family of functions from topological spaces into is the finest topology on that makes those functions continuous.
In topology and related fields of mathematics, a sequential space is a topological space that satisfies a very weak axiom of countability.
In topology and other branches of mathematics, a topological space X is locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets.
In the mathematical field of topology, a hyperconnected space or irreducible space is a topological space X that cannot be written as the union of two proper closed sets. The name irreducible space is preferred in algebraic geometry.
In the field of topology, a Fréchet–Urysohn space is a topological space with the property that for every subset the closure of in is identical to the sequential closure of in Fréchet–Urysohn spaces are a special type of sequential space.
In mathematics, a polyadic space is a topological space that is the image under a continuous function of a topological power of an Alexandroff one-point compactification of a discrete space.
In topology, a subfield of mathematics, filters are special families of subsets of a set that can be used to study topological spaces and define all basic topological notions such a convergence, continuity, compactness, and more. Filters also provide a common framework for defining various types of limits of functions such as limits from the left/right, to infinity, to a point or a set, and many others. Special types of filters called ultrafilters have many useful technical properties and they may often be used in place of arbitrary filters.
In functional analysis and related areas of mathematics, a complete topological vector space is a topological vector space (TVS) with the property that whenever points get progressively closer to each other, then there exists some point towards which they all get closer to. The notion of "points that get progressively closer" is made rigorous by Cauchy nets or Cauchy filters, which are generalizations of Cauchy sequences, while "point towards which they all get closer to" means that this net or filter converges to Unlike the notion of completeness for metric spaces, which it generalizes, the notion of completeness for TVSs does not depend on any metric and is defined for all TVSs, including those that are not metrizable or Hausdorff.
In mathematics, an LB-space, also written (LB)-space, is a topological vector space that is a locally convex inductive limit of a countable inductive system of Banach spaces. This means that is a direct limit of a direct system in the category of locally convex topological vector spaces and each is a Banach space.
In mathematics, specifically topology, a sequence covering map is any of a class of maps between topological spaces whose definitions all somehow relate sequences in the codomain with sequences in the domain. Examples include sequentially quotient maps, sequence coverings, 1-sequence coverings, and 2-sequence coverings. These classes of maps are closely related to sequential spaces. If the domain and/or codomain have certain additional topological properties then these definitions become equivalent to other well-known classes of maps, such as open maps or quotient maps, for example. In these situations, characterizations of such properties in terms of convergent sequences might provide benefits similar to those provided by, say for instance, the characterization of continuity in terms of sequential continuity or the characterization of compactness in terms of sequential compactness.