Four examples and two non-examples of topologies on the three-point set {1,2,3}. The bottom-left example is not a topology because the union {2,3} of {2} and {3} is missing; the bottom-right example is not a topology because the intersection {2} of {1,2} and {2,3} is missing.

Topological spaces are mathematical structures that allow the formal definition of such concepts such as convergence, connectedness and continuity. 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 In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function They appear in virtually every branch of modern mathematics and are a central unifying notion. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The branch of mathematics that studies topological spaces in their own right is called topology. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of

Definition

A topological space is a set X together with T, a collection of subsets of X, satisfying the following axioms:

1. The empty set and X are in T. In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members
2. The union of any collection of sets in T is also in T. 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
3. The intersection of any finite collection of sets in T is also in T. 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

The collection T is called a topology on X. The elements of X are usually called points, though they can be any mathematical objects. A topological space in which the points are functions is called a function space. The sets in T are the open sets, and their complements in X are called closed sets. In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in 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 Topology and related branches of Mathematics, a closed set is a set whose complement is open. Some sets are neither open nor closed. Other sets are both open and closed. Some sets are open but not closed, some are closed but not open.

Examples

1. X={1,2,3,4} and collection T={{},{1,2,3,4}} of two subsets of X form a trivial topology.
2. X={1,2,3,4} and collection T={{},{2},{1,2},{2,3},{1,2,3},{1,2,3,4}} of six subsets of X form another topology.
3. X=Z, the set of integers and collection T equal to all finite subsets of the integers plus Z itself is not a topology, because (for example) the union over all finite sets not containing zero is infinite but is not all of Z, and so is not in T.

Equivalent definitions

There are many other equivalent ways to define a topological space. In Mathematics, a Topological space is usually defined in terms of Open sets However there are many equivalent characterizations of the Category (In other words, each of the following defines a category equivalent to the category of topological spaces above. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets ) For example, using de Morgan's laws, the axioms defining open sets above become axioms defining closed sets:

1. The empty set and X are closed. In Logic, De Morgan's laws or De Morgan's theorem are rules in Formal logic relating pairs of dual Logical operators in a systematic manner expressed
2. The intersection of any collection of closed sets is also closed.
3. The union of any pair of closed sets is also closed.

Using these axioms, another way to define a topological space is as a set X together with a collection T of subsets of X satisfying the following axioms:

1. The empty set and X are in T.
2. The intersection of any collection of sets in T is also in T.
3. The union of any pair of sets in T is also in T.

Under this definition, the sets in the topology T are the closed sets, and their complements in X are the open sets.

Another way to define a topological space is by 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 In Mathematics, an operator is a function which operates on (or modifies another function In Mathematics, given a set S, the power set (or powerset) of S, written \mathcal{P}(S P ( S)

A neighbourhood of a point x is any set that contains an open set containing x. In Topology and related areas of Mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a Topological space. 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.

A net is a generalisation of the concept of sequence. This article is about nets in Topological spaces and not about ε-nets in Metric spaces In Topology and related areas of Mathematics In Mathematics, a sequence is an ordered list of objects (or events A topology is completely determined if for every net in X the set of its accumulation points is specified. This is a glossary of some terms used in the branch of Mathematics known as Topology.

Comparison of topologies

Main article: Comparison of topologies

A variety of topologies can be placed on a set to form a topological space. In Topology and related areas of Mathematics comparison of topologies refers to the fact that two Topological structures on a given set may stand in relation When every set in a topology T₁ is also in a topology T₂, we say that T₂ is finer than T₁, and T₁ is coarser than T₂. In Topology and related areas of Mathematics comparison of topologies refers to the fact that two Topological structures on a given set may stand in relation In Topology and related areas of Mathematics comparison of topologies refers to the fact that two Topological structures on a given set may stand in relation 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.

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 elements of

Continuous functions

A function between topological spaces is said to be continuous if the inverse image of every open set is open. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage 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. Topological equivalence redirects here see also Topological equivalence (dynamical systems. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property In Mathematics, if &fnof is a function from A to B then an inverse function for &fnof is a function in the opposite direction from B Two spaces are said to be homeomorphic if there exists a homeomorphism between them. From the standpoint of topology, homeomorphic spaces are essentially identical.

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. In Mathematics, the category of topological spaces, often denoted Top, is the category whose objects are Topological spaces and whose In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships 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 K-theory, to name just a few. In Mathematics, an invariant is something that does not change under a set of transformations The property of being an invariant is invariance. In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical In Mathematics (especially Algebraic topology and Abstract algebra) homology (in Greek ὁμός homos "identical" is In Mathematics, K-theory is a tool used in several disciplines

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. In Topology, a discrete space is a particularly simple example of a Topological space or similar structure one in which the points are " isolated " 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. In Topology, a Topological space with the trivial topology is one where the only Open sets are the Empty set and the entire space 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. However, oftentimes topological spaces are required to be Hausdorff spaces where limit points are unique. In Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological space

There are many ways of defining a topology on R, the set of real numbers. In Mathematics, the real numbers may be described informally in several different ways The standard topology on R is generated by the open intervals. The set of all open intervals forms a base or basis for the topology, meaning that every open set is a union of some collection of sets from the base. In Mathematics, a base (or basis) B for a Topological space X with topology T is a collection of Open sets In particular, this means that a set is open if there exists an open interval of non zero radius about every point in the set. More generally, the Euclidean spaces Rⁿ can be given a topology. In the usual topology on Rⁿ the basic open sets are the open balls. In Mathematics, a ball is the inside of a Sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions and for metric Similarly, C and Cⁿ have a standard topology in which the basic open sets are open balls.

Every metric space can be given a metric topology, in which the basic open sets are open balls defined by the metric. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined This is the standard topology on any normed vector space. In Mathematics, with 2- or 3-dimensional vectors with real -valued entries the idea of the "length" of a vector is intuitive and can easily be extended to On a finite-dimensional vector space this topology is the same for all norms.

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 which operates on (or modifies another function For functional analysis as used in psychology see the Functional analysis (psychology article

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 that is a Locally compact Topological field with respect to a non-discrete topology

Every manifold has a natural topology since it is locally Euclidean. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be Similarly, every simplex and every simplicial complex inherits a natural topology from Rⁿ. In Geometry, a simplex (plural simplexes or simplices) or n -simplex is an n -dimensional analogue of a triangle In Mathematics, a simplicial complex is a Topological space of a particular kind constructed by "gluing together" points Line segments

The Zariski topology is defined algebraically on the spectrum of a ring or an algebraic variety. In Mathematics, namely Algebraic geometry, the Zariski topology is a particular Topology chosen for algebraic varieties that reflects the algebraic In Abstract algebra and Algebraic geometry, the spectrum of a Commutative ring R, denoted by Spec( R) is defined to be the set of This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety On Rⁿ or Cⁿ, the closed sets of the Zariski topology are the solution sets of systems of polynomial equations. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations

A linear graph has a natural topology that generalises many of the geometric aspects of graphs with vertices and edges. In Mathematics and Computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects

Sierpiński space is the simplest non-trivial, non-discrete topological space. In Mathematics, Sierpiński space (or the connected two-point set) is a Finite topological space with two points only one of which is closed It has important relations to the theory of computation and semantics.

There exist numerous topologies on any given finite set. In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2. Such spaces are called finite topological spaces. In Mathematics, a finite topological space is a Topological space for which the underlying point set is finite. Finite spaces are often used to provide examples or counterexamples to conjectures about topological spaces in general.

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. In Mathematics, a cofinite Subset of a set X is a subset Y whose complement in X is a finite set This is the smallest T₁ topology on any infinite set. In Topology and related branches of Mathematics, T1 spaces and R0 spaces are particular kinds of Topological spaces The

An uncountable set can be given the cocountable topology, in which a set is defined to be open if it is either empty or its complement is countable. The cocountable topology or countable complement topology on any set X consists of the Empty set and all Cocountable subsets of X, This topology serves as a useful counterexample in many situations.

The real line can also be given the lower limit topology. In Mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set R of Real numbers; 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.

If Γ is an ordinal number, then the set Γ = [0, Γ) may be endowed with the order topology generated by the intervals (a, b), [0, b) and (a, Γ) where a and b are elements of Γ. In Set theory, an ordinal number, or just ordinal, is the Order type of a Well-ordered set. In Mathematics, an order topology is a certain 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. In Topology and related areas of Mathematics, a subspace of a Topological space X is a Subset S of X which is For any indexed family 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. In Mathematics, an indexed family of sets is defined in stages beginning with the more general concept of an indexed family of elements, which is really just an alternative In Topology and related areas of Mathematics, a product space is the Cartesian product of a family of Topological spaces equipped with a natural In Mathematics, a projection is any one of several different types of functions mappings operations or transformations for example the following In 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.

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 Topology and related areas of Mathematics, a quotient space (also called an identification space) is intuitively speaking the result of identifying In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage 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. In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" The map f is then the natural projection onto the set of equivalence classes. In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X

The Vietoris topology on the set of all non-empty subsets of a topological space X, named for Leopold Vietoris, is generated by the following basis: for every n-tuple U₁, . Leopold Vietoris ( Radkersburg, June 4, 1891 - Innsbruck, April 9, 2002) was an Austrian Mathematician . . , 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 non-empty intersection with each U_i.

Classification of topological spaces

Topological spaces can be broadly classified, up to homeomorphism, by their topological properties. 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 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. In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of 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

Topological spaces with algebraic structure

For any algebraic objects we can introduce the discrete topology, under which the algebraic operations are continuous functions. In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, 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 Mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the 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 In Mathematics, a local field is a special type of field that is a Locally compact Topological field with respect to a non-discrete topology

Topological spaces with order structure

• Spectral. A space is spectral if and only if it is the prime spectrum of a ring (Hochster theorem). In Mathematics, a Topological space X with Topology Ω is said to be spectral if 1 X is Compact and T0 In Abstract algebra and Algebraic geometry, the spectrum of a Commutative ring R, denoted by Spec( R) is defined to be the set of Melvin Hochster (born August 2 1943 is an eminent American Mathematician, regarded as one of the leading Commutative algebraists active today

• Specialization preorder. In a space the specialization (or canonical) preorder is defined by x ≤ y if and only if cl{x} ⊆ cl{y}. In the branch of Mathematics known as Topology, the specialization (or canonical) preorder is a natural Preorder on the set of the 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 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

Specializations and generalizations

The following spaces and algebras are either more specialized or more general than the topological spaces discussed above.

• Proximity spaces provide a notion of closeness of two sets. In Topology, a proximity space is an axiomatization of notions of "nearness" that hold set-to-set as opposed to the better known point-to-set notions that characterize
• Metric spaces embody a metric, a precise notion of distance between points. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined
• Uniform spaces axiomatize ordering the distance between distinct points. In the Mathematical field of Topology, a uniform space is a set with a uniform structure.
• Cauchy spaces axiomatize the ability to test whether a net is Cauchy. In General topology and analysis, a Cauchy space is a generalization of Metric spaces and Uniform spaces for which the notion of Cauchy convergence In Mathematics, a Cauchy net generalizes the notion of Cauchy sequence to nets defined on Uniform spaces A net ( x α Cauchy spaces provide a general setting for studying completions. In Mathematical analysis, a Metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has
• Convergence spaces capture some of the features of convergence of filters. In Mathematics, a filter is a special Subset of a Partially ordered set.
• σ-algebras build on the notion of measurable sets. In Mathematics, a σ-algebra (or sigma-algebra) ( Sigma is a Greek letter upper case Σ lower case σ over a set X is a nonempty In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with

See also

• T0 space
• T1 space
• Hausdorff space (T2)
• Completely Hausdorff space
• Urysohn space
• T3 space
• Tychonoff space
• Normal Hausdorff space (T4)
• Completely normal Hausdorff space (T5)
• Perfectly normal Hausdorff space (T6)

References

• Armstrong, M. In Topology and related branches of Mathematics, the T0 spaces or Kolmogorov spaces, named after Andrey Kolmogorov, form a broad class In Topology and related branches of Mathematics, T1 spaces and R0 spaces are particular kinds of Topological spaces The In Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological space In Topology, completely Hausdorff spaces and Urysohn (or T2½) spaces are types of Topological spaces satisfying slightly In Topology, completely Hausdorff spaces and Urysohn (or T2½) spaces are types of Topological spaces satisfying slightly In Topology and related fields of Mathematics, regular spaces and T3 spaces are particularly convenient kinds of Topological spaces In Topology and related branches of Mathematics, Tychonoff spaces and completely regular spaces are particularly nice kinds of Topological spaces In Topology and related branches of Mathematics, normal spaces, T4 spaces, T5 spaces, and T6 spaces In Topology and related branches of Mathematics, normal spaces, T4 spaces, T5 spaces, and T6 spaces In Topology and related branches of Mathematics, normal spaces, T4 spaces, T5 spaces, and T6 spaces A. ; Basic Topology, Springer; 1st edition (May 1, 1997). ISBN 0-387-90839-0.
• Bredon, Glen E. , Topology and Geometry (Graduate Texts in Mathematics), Springer; 1st edition (October 17, 1997). ISBN 0-387-97926-3.
• Bourbaki, Nicolas; Elements of Mathematics: General Topology, Addison-Wesley (1966). Nicolas Bourbaki is the collective Pseudonym under which a group of (mainly French) 20th-century Mathematicians wrote a series of books presenting an exposition
• Čech, Eduard; Point Sets, Academic Press (1969). Eduard Čech (ˈʧɛx ( June 29, 1893 – March 15, 1960) was a Czech Mathematician born in Stračov, Bohemia
• Fulton, William, Algebraic Topology, (Graduate Texts in Mathematics), Springer; 1st edition (September 5, 1997). ISBN 0-387-94327-7.
• Lipschutz, Seymour; Schaum's Outline of General Topology, McGraw-Hill; 1st edition (June 1, 1968). ISBN 0-07-037988-2.
• Munkres, James; Topology, Prentice Hall; 2nd edition (December 28, 1999). ISBN 0-13-181629-2.
• Runde, Volker; A Taste of Topology (Universitext), Springer; 1st edition (July 6, 2005). ISBN 0-387-25790-X.
• Steen, Lynn A. and Seeback, J. Arthur Jr. ; Counterexamples in Topology, Holt, Rinehart and Winston (1970). Counterexamples in Topology (1970 2nd ed 1978 is a book on Mathematics by topologists Lynn Steen and J ISBN 0-03-079485-4.
• Wilard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.  

External links

• Topological space on PlanetMath