Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion. The branch of mathematics that studies topological spaces in their own right is called topology. In the absence of a more specific context, convergence denotes the approach toward a definite value, as time goes on; or to a definite point, a common view or opinion, or toward a fixed or equilibrium state. ...
Connected and disconnected subspaces of RÂ². The space A at top is connected; the shaded space B at bottom is not. ...
In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ...
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. ...
A MÃ¶bius strip, a surface with only one side and one edge; such shapes are an object of study in topology. ...
This article is technical. For a general overview of the subject, see the article on topology. A MÃ¶bius strip, a surface with only one side and one edge; such shapes are an object of study in topology. ...
Definition
A topological space is a set X together with a collection T of subsets of X satisfying the following axioms: In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ...
A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ...
This article does not cite its references or sources. ...
 The empty set and X are in T.
 The union of any collection of sets in T is also in T.
 The intersection of any pair of sets in T is also in T.
The collection T is a topology on X, and the elements of X are called points. The sets in T are the open sets, and their complements in X are the closed sets. The requirement that the union of any collection of open sets be open is more stringent than simply requiring that all pairwise unions be open, as the former includes unions of infinite collections of sets. It follows that a closed set must satisfy the following: In mathematics and more specifically set theory, the empty set is the unique set which contains no 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, 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 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...
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 branches of mathematics, a closed set is a set whose complement is open. ...
 The empty set and X are closed (as well as being open).
 The intersection of any collection of closed sets is also closed
 The union of any pair of closed sets is also closed
By induction, the intersection of any finite collection of open sets is open. Thus, since the union of the empty collection is the empty set, and the intersection of the empty collection is (by convention) X, an equivalent definition can be given by requiring that a topology be closed under unions and finite intersections. In mathematics and more specifically set theory, the empty set is the unique set which contains no elements. ...
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. ...
Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. ...
Comparison of topologies 
A variety of topologies can be placed on a set to form a topological space. When every set in a topology T_{1} is also in a topology T_{2}, we say that T_{2} is finer than T_{1}, and T_{1} is coarser than T_{2}. A proof which relies only on the existence of certain open sets will also hold for any finer topology, and similarly a proof that relies only on certain sets not being open applies to any coarser topology. The terms larger and smaller are sometimes used in place of finer and coarser, respectively. The terms stronger and weaker are also used in the literature, but with little agreement on the meaning, so one should always be sure of an author's convention when reading. In topology and related areas of mathematics comparison of topologies refers to the fact that two topological structures on a given set X may stand in relation to each other. ...
In mathematics, the possible topologies on a given set X form a partially ordered set: if a collection τ1 of subsets of X contains each subset in a collection τ2, and these are both topologies on X, we say that τ1 is a finer (alt. ...
In mathematics, the possible topologies on a given set X form a partially ordered set: if a collection τ1 of subsets of X contains each subset in a collection τ2, and these are both topologies on X, we say that τ1 is a finer (alt. ...
The collection of all topologies on a given fixed set X forms a complete lattice: if F = {T_{α} : α in A} is a collection of topologies on X, then the meet of F is the intersection of F, and the join of F is the meet of the collection of all topologies on X which contain every member of F. In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). ...
In mathematics the infimum of a subset of some set is the greatest element, not necessarily in the subset, that is less than or equal to all other elements of the subset. ...
In mathematics, the supremum of an ordered set S is the least element that is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound (also lub and LUB). ...
Continuous functions A function between topological spaces is said to be continuous if the inverse image of every open set is open. This is an attempt to capture the intuition that there are no "breaks" or "separations" in the function. A homeomorphism is a bijection that is continuous and whose inverse is also continuous. Two spaces are said to be homeomorphic if there exists a homeomorphism between them. From the standpoint of topology, homeomorphic spaces are essentially identical. Partial plot of a function f. ...
In topology, a continuous function is generally defined as one for which preimages of open sets are open. ...
In mathematics, the image of an element x in a set X under the function f : X → Y, denoted by f(x), is the unique y in Y that is associated with x. ...
In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ...
A bijective function. ...
In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ...
In category theory, Top, the category of topological spaces with topological spaces as objects and continuous functions as morphisms is one of the fundamental categories in mathematics. The attempt to classify the objects of this category (up to homeomorphism) by invariants has motivated and generated entire areas of research, such as homotopy theory, homology theory, and Ktheory, to name just a few. The category Top has topological spaces as objects and continuous maps as morphisms. ...
Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ...
In mathematics, a morphism is an abstraction of a structurepreserving process between two mathematical structures. ...
In mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. ...
In mathematics, an invariant is something that does not change under a set of transformations. ...
The two bold paths shown above are homotopic relative to their endpoints. ...
In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homos = identical) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ...
In mathematics, Ktheory is, firstly, an extraordinary cohomology theory which consists of topological Ktheory. ...
Equivalent definitions There are many other equivalent ways to define a topological space. (In other words, each of the following defines a category equivalent to the category of topological spaces above.) For example, using de Morgan's laws, the axioms defining open sets become axioms defining closed sets: In mathematics, a topological space is usually defined in terms of open sets. ...
In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ...
note that demorgans laws are also a big part in circut design. ...
 The empty set and X are closed.
 The intersection of any collection of closed sets is also closed.
 The union of any pair of closed sets is also closed.
Another way to define a topological space is using the Kuratowski closure axioms, which define the closed sets as the fixed points of an operator on the power set of X. In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms which can be used to define a topological structure on a set. ...
In mathematics, an operator is a function that performs some sort of operation on a number, variable, or function. ...
In mathematics, given a set S, the power set (or powerset) of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ...
A neighbourhood of a point x is any set that contains an open set containing x. The neighbourhood system at x consists of all neighbourhoods of x. A topology can be determined by a set of axioms concerning all neighbourhood systems. This is a glossary of some terms used in the branch of mathematics known as topology. ...
A net is a generalisation of the concept of sequence. A topology is completely determined if for every net in X the set of its accumulation points is specified. In topology and related areas of mathematics a net or MooreSmith sequence is a generalization of a sequence, intended to unify the various notions of limit and generalize them to arbitrary topological spaces. ...
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. ...
This is a glossary of some terms used in the branch of mathematics known as topology. ...
Examples of topological spaces A given set may have many different topologies. If a set is given a different topology, it is viewed as a different topological space. Any set can be given the discrete topology in which every set is open. The only convergent sequences or nets in this topology are those that are eventually constant. Also, any set can be given the trivial topology (also called the indiscrete topology), in which only the empty set and the whole space are open. Every sequence and net in this topology converges to every point of the space. This example shows that in general topological spaces, limits of sequences need not be unique. In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are isolated from each other in a certain sense. ...
In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. ...
There are many ways of defining a topology on R, the set of real numbers. The standard topology on R is generated by the open intervals. The open intervals form a base or basis for the topology, meaning that every open set is a union of basic open sets. More generally, the Euclidean spaces R^{n} can be given a topology. In the usual topology on R^{n} the basic open sets are the open balls. Similarly, C and C^{n} have a standard topology in which the basic open sets are open balls. In mathematics, the real numbers may be described informally in several different ways. ...
In mathematics, a base (or basis) B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B. We say that the base generates the topology T. Bases...
Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...
The solid interior of a sphere or circle; in mathematics, latter terms refer specifically to the (n1)dimensional surface of an ndimensional solid ball. ...
Every metric space can be given a metric topology, in which the basic open sets are open balls defined by the metric. This is the standard topology on any normed vector space. In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ...
In mathematics, with 2 or 3dimensional vectors with realvalued entries, the idea of the length of a vector is intuitive and can easily be extended to any real vector space Rn. ...
Many sets of operators in functional analysis are endowed with topologies that are defined by specifying when a particular sequence of functions converges to the zero function. In mathematics, an operator is a function that performs some sort of operation on a number, variable, or function. ...
Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ...
Any local field has a topology native to it, and this can be extended to vector spaces over that field. In mathematics, a local field is a special type of field which has the additional property that it is a complete metric space with respect to a discrete valuation. ...
Every manifold has a natural topology since it is locally Euclidean. Similarly, every simplex and every simplicial complex inherits a natural topology from R^{n}. On a sphere, the sum of the angles of a triangle is not equal to 180Â°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ...
In geometry, a simplex (plural: simplices) or nsimplex is an ndimensional analogue of a triangle. ...
In mathematics, a simplicial complex is a topological space of a particular kind, built up of points, line segments, triangles, and their ndimensional counterparts. ...
The Zariski topology is defined algebraically on the spectrum of a ring or an algebraic variety. On R^{n} or C^{n}, the closed sets of the Zariski topology are the solution sets of systems of polynomial equations. This article needs to be cleaned up to conform to a higher standard of quality. ...
In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec(R), is defined to be the set of all proper prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. ...
In classical algebraic geometry (and to some extent also in modern algebraic geometry), the main objects of study are algebraic varieties. ...
In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ...
A linear graph has a natural topology that generalises many of the geometric aspects of graphs with vertices and edges. Sierpiński space is the simplest nontrivial, nondiscrete topological space. It has important relations to the theory of computation and semantics. This is a topology where inclusion of a particular point defines openness. ...
Any infinite set can be given the cofinite topology in which the open sets are the empty set and the sets whose complement is finite. This is the smallest T_{1} topology on any infinite set. In mathematics, a cofinite subset of a set X is a subset Y whose complement in X is a finite set. ...
The title given to this article is incorrect due to technical limitations. ...
The real line can also be given the lower limit topology. Here, the basic open sets are the half open intervals [a, b). This topology on R is strictly finer than the Euclidean topology defined above; a sequence converges to a point in this topology if and only if it converges from above in the Euclidean topology. This example shows that a set may have many distinct topologies defined on it. In mathematics, the lower limit topology or right halfopen interval topology is a topology defined on the set R of real numbers; it is different from the standard topology on R and has a number of interesting properties. ...
If Γ is an ordinal number, then the set [0, Γ] may be endowed with the order topology generated by the intervals (a, b), where a and b are elements of Γ. Ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ...
In mathematics, the order topology is a topology that can be defined on any totally ordered set. ...
Topological constructions Every subset of a topological space can be given the subspace topology in which the open sets are the intersections of the open sets of the larger space with the subset. For any nonempty collection of topological spaces, the product can be given the product topology, which is generated by the inverse images of open sets of the factors under the projection mappings. For example, in finite products, a basis for the product topology consists of all products of open sets. For infinite products, there is the additional requirement that in a basic open set, all but finitely many of its projections are the entire space. 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 natural topology induced from that of X called the subspace topology (or the relative topology, or the induced topology). ...
In topology, the cartesian product of topological spaces is turned into a topological space in the following way. ...
In mathematics, a projection is any one of several different types of functions, mappings, operations, or transformations, for example, the following: A settheoretic operation typified by the jth projection map, written , that takes an element of the cartesian product to the value . ...
A quotient space is defined as follows: if X is a topological space and Y is a set, and if f : X → Y is a surjective function, then the quotient topology on Y is the collection of subsets of Y that have open inverse images under f. In other words, the quotient topology is the finest topology on Y for which f is continuous. A common example of a quotient topology is when an equivalence relation is defined on the topological space X. The map f is then the natural projection onto the set of equivalence classes. In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or gluing together certain points of a given space. ...
A surjective function. ...
Partial plot of a function f. ...
In mathematics, the image of an element x in a set X under the function f : X → Y, denoted by f(x), is the unique y in Y that is associated with x. ...
In mathematics, an equivalence relation, denoted by an infix ~, is a binary relation on a set X that is reflexive, symmetric, and transitive. ...
In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x âˆˆ X  x ~ a } The notion of equivalence classes is useful for constructing sets out...
The Vietoris topology on the set of all nonempty subsets of a topological space X, named for Leopold Vietoris, is generated by the following basis: for every ntuple U_{1}, ..., U_{n} of open sets in X, we construct a basis set consisting of all subsets of the union of the U_{i} which have nonempty intersection with each U_{i}. Dr. Leopold Vietoris (June 4, 1891  April 9, 2002) was an Austrian mathematician who gained added fame by reaching an extremely old age. ...
Classification of topological spaces Topological spaces can be broadly classified, up to homeomorphism, by their topological properties. A topological property is a property of spaces that is invariant under homeomorphisms. To prove that two spaces are not homeomorphic it is sufficient to find a topological property which is not shared by them. Examples of such properties include connectedness, compactness, and various separation axioms. Look up Up to on Wiktionary, the free dictionary In mathematics, the phrase up to xxxx indicates that members of an equivalence class are to be regarded as a single entity for some purpose. ...
In topology and related areas of mathematics a topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms. ...
In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ...
In mathematics, a compact space is a space that resembles a closed and bounded subset of Euclidean space Rn in that it is small in a certain sense and contains all its limit points. The modern general definition calls a topological space compact if every open cover of it has...
In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. ...
See the article on topological properties for more details and examples. In topology and related areas of mathematics a topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms. ...
Topological spaces with algebraic structure For any algebraic objects we can introduce the discrete topology, under which the algebraic operations are continuous functions. For any such structure which is not finite, we often have a natural topology which is compatible with the algebraic operations in the sense that the algebraic operations are still continuous. This leads to concepts such as topological groups, topological vector spaces, topological rings and local fields. In universal algebra, a branch of pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, satisfying some axioms. ...
In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G Ã— G â†’ G and the inverse operation G â†’ G are continuous maps. ...
In mathematics a topological vector space is one of the basic structures investigated in functional analysis. ...
In mathematics, a topological ring is a ring R which is also a topological space such that both the addition and the multiplication are continuous as maps R × R → R, where R × R carries the product topology. ...
In mathematics, a local field is a special type of field which has the additional property that it is a complete metric space with respect to a discrete valuation. ...
Topological spaces with order structure In mathematics, a topological space X is said to be spectral if 1) X is compact and T0; 2) The set C(X) of all compactopen subsets of (X,Ω) is a sublattice of Ω and a base for the topology; 3) X is sober, that is any nonempty...
In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec(R), is defined to be the set of all proper prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. ...
Melvin Hochster is a leading American mathematician, regarded as perhaps the leading practioner in the field of commutative algebra, which provides the setting for such highly applicable notions as GrÃ¶bner bases. ...
In the branch of mathematics known as topology the specialization (or canonical) preorder defines a preorder on the set of the points of a topological space. ...
In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms which can be used to define a topological structure on a set. ...
In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms which can be used to define a topological structure on a set. ...
Specializations and generalizations Topological spaces provide the most common notions of closeness and convergence for a space, but it may be possible in some cases to study more specialized or more general notions.  Proximity spaces provide a notion of closeness of two sets.
 Metric spaces have a precise notion of distance between points, so that the closeness of any disparate pair of points can be compared.
 Uniform spaces carry a structure which axiomatize the idea of comparing the closeness of disparate pairs of points.
 Cauchy spaces carry a structure which axiomatize the ability to test whether a net is Cauchy. Cauchy spaces provide a general setting for studying completions.
 Convergence spaces carry a structure which captures some of the features of convergence of filters.
 σalgebras provide a selection of sets whose size may be measured
In topology, a proximity space is an axiomatization of notions of nearness that hold settoset, as opposed to the better known pointtoset notions that characterize topological spaces. ...
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ...
In the mathematical field of topology, a uniform space is a set with a uniform structure. ...
In general topology, a Cauchy space is a structure introduced by H. H. Keller in 1968, as an axiomatic tool derived from the idea of a Cauchy filter, in order to study completeness in topological spaces. ...
In mathematical analysis, a Cauchy sequence is a sequence whose terms become arbitrarily close to each other as the sequence progresses. ...
In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For...
In mathematics, a Ïƒalgebra (pronounced sigmaalgebra) or Ïƒfield over a set X is a collection Î£ of subsets of X that is closed under countable set operations, meaning that the union or the intersection of countably many members of the algebra is also a member. ...
In mathematics, a measure is a function that assigns a number, e. ...
References  Armstrong, M. A.; Basic Topology, Springer; 1st edition (May 1, 1997). ISBN 0387908390.
 Bredon, Glen E., Topology and Geometry (Graduate Texts in Mathematics), Springer; 1 edition (October 17, 1997). ISBN 0387979263.
 Bourbaki, Nicolas; Elements of Mathematics: General Topology, AddisonWesley (1966).
 Čech, Eduard; Point Sets, Academic Press (1969).
 Fulton, William, Algebraic Topology, (Graduate Texts in Mathematics), Springer; 1 edition (September 5, 1997). ISBN 0387943277.
 Lipschutz, Seymour; Schaum's Outline of General Topology, McGrawHill; 1st edition (June 1, 1968). ISBN 0070379882.
 Munkres, James; Topology, Prentice Hall; 2nd edition (December 28, 1999). ISBN 0131816292.
 Runde, Volker; A Taste of Topology (Universitext), Springer; 1st edition (July 6, 2005). ISBN 038725790X.
 Steen, Lynn A. and Seeback, J. Arthur Jr.; Counterexamples in Topology, Holt, Rinehart and Winston (1970). ISBN 0030794854.
 Wilard, Stephen (2004). General Topology. Dover Publications. ISBN 0486434796.
Nicolas Bourbaki is the collective allonym under which a group of mainly French 20thcentury mathematicians wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. ...
Wikipedia does not have an article with this exact name. ...
Counterexamples in Topology (1970) is a mathematics book by topologists Lynn A. Steen and J. Arthur Seebach, Jr. ...
External links  Topological Space, Open Sets, Closed Sets, Interior, Closure definitions and basic theorems.
 Topological Space at ProvenMath strict definitions and formal proofs.
