- Topological space
**Topological spaces**are mathematical structures that allow the formal definition of concepts such asconvergence , connectedness, and continuity. They appear in virtually every branch of modernmathematics and are a central unifying notion. The branch of mathematics that studies topological spaces in their own right is calledtopology .**Definition**A

**topological space**is a set "X" together with "T", a collection ofsubset s of "X", satisfying the followingaxiom s:# 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 finite collection of sets in "T" is also in "T".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, and their complements in "X" are calledopen set s. A set may be neither closed nor open, either closed or open, or both.closed set s**Examples**# "X" = {1, 2, 3, 4} and collection "T" = }, {1, 2, 3, 4 of two subsets of "X" form a trivial topology.

# "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.

# "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 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 above become axioms defining closed sets:# 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.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:

# The empty set and "X" are in "T".

# The intersection of any collection of sets in "T" is also in "T".

# 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 anoperator on thepower set of`X`.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.

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.**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.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".**Continuous functions**A function between topological spaces is said to be

**continuous**if theinverse 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. Ahomeomorphism is abijection 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.In category theory,

**Top**, thecategory of topological spaces with topological spaces as objects and continuous functions asmorphism s 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, andK-theory , to name just a few.**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 subset 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. However, oftentimes topological spaces are required to be Hausdorff spaces where limit points are unique.There are many ways of defining a topology on

**R**, the set ofreal number s. 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 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, theEuclidean space s**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.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 anynormed vector space . On a finite-dimensional vector space this topology is the same for all norms.Many sets of

operator s infunctional analysis are endowed with topologies that are defined by specifying when a particular sequence of functions converges to the zero function.Any

local field has a topology native to it, and this can be extended to vector spaces over that field.Every

manifold has a natural topology since it is locally Euclidean. Similarly, everysimplex and everysimplicial complex inherits a natural topology from**R**^{n}.The

Zariski topology is defined algebraically on thespectrum of a ring or analgebraic variety . On**R**^{"n"}or**C**^{"n"}, the closed sets of the Zariski topology are the solution sets of systems ofpolynomial equations.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 non-trivial, non-discrete topological space. It has important relations to the theory of computation and semantics.There exist numerous topologies on any given

finite set . Such spaces are calledfinite topological space s. 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. This is the smallest T_{1}topology on any infinite set.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. This topology serves as a useful counterexample in many situations.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.If Γ is an

ordinal number , then the set Γ = [0, Γ) may be endowed with theorder topology generated by the intervals ("a", "b"), [0, "b") and ("a", Γ) where "a" and "b" are elements of Γ.**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 anyindexed family of topological spaces, the product can be given theproduct 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.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 openinverse image s 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 anequivalence relation is defined on the topological space "X". The map "f" is then the natural projection onto the set ofequivalence class es.The

**Vietoris topology**on the set of all non-empty subsets of a topological space "X", named forLeopold Vietoris , is generated by the following basis: for every "n"-tuple "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 non-empty intersection with each "U"_{"i"}.**Classification of topological spaces**Topological spaces can be broadly classified,

up to homeomorphism, by theirtopological 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 variousseparation axiom s.See the article on "

topological properties " for more details and examples.**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 group s,topological vector space s,topological ring s andlocal field s.**Topological spaces with order structure***

**Spectral**. A space is spectral if and only if it is the primespectrum of a ring (Hochster theorem).*

**Specialization preorder**. In a space the**specialization**(or**canonical**)**preorder**is defined by "x" ≤ "y" if and only if cl{"x"} ⊆ cl{"y"}.**Specializations and generalizations**The following spaces and algebras are either more specialized or more general than the topological spaces discussed above.

*Proximity space s provide a notion of closeness of two sets.

*Metric space s embody a metric, a precise notion of distance between points.

*Uniform space s axiomatize ordering the distance between distinct points.

*Cauchy space s axiomatize the ability to test whether a net is Cauchy. Cauchy spaces provide a general setting for studying completions.

*Convergence space s capture some of the features of convergence of filters.

*σ-algebra s build on the notion of measurable sets.**ee 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. 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).

* ech, Eduard; "Point Sets", Academic Press (1969).

* 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 Seebach, J. Arthur Jr.; "Counterexamples in Topology ", Holt, Rinehart and Winston (1970). ISBN 0-03-079485-4.

*cite book | author=Willard, Stephen | title=General Topology | publisher=Dover Publications | year=2004 | id=ISBN 0-486-43479-6**External links***planetmath reference|id=380|title=Topological space

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