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OR:

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets, which is easier than the others to manipulate. A topological space is the most general type of a mathematical space that allows for the definition of limits, continuity, and connectedness. Common types of topological spaces include
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
s, metric spaces and manifolds. Although very general, the concept of topological spaces is fundamental, and used in virtually every branch of modern mathematics. The study of topological spaces in their own right is called point-set topology or general topology.

# History

Around 1735, Leonhard Euler discovered the formula $V - E + F = 2$ relating the number of vertices, edges and faces of a convex polyhedron, and hence of a planar graph. The study and generalization of this formula, specifically by Cauchy (1789-1857) and L'Huilier (1750-1840), boosted the study of topology. In 1827, Carl Friedrich Gauss published ''General investigations of curved surfaces'', which in section 3 defines the curved surface in a similar manner to the modern topological understanding: "A curved surface is said to possess continuous curvature at one of its points A, if the direction of all the straight lines drawn from A to points of the surface at an infinitely small distance from A are deflected infinitely little from one and the same plane passing through A." Yet, "until Riemann's work in the early 1850s, surfaces were always dealt with from a local point of view (as parametric surfaces) and topological issues were never considered". " Möbius and Jordan seem to be the first to realize that the main problem about the topology of (compact) surfaces is to find invariants (preferably numerical) to decide the equivalence of surfaces, that is, to decide whether two surfaces are homeomorphic or not." The subject is clearly defined by Felix Klein in his " Erlangen Program" (1872): the geometry invariants of arbitrary continuous transformation, a kind of geometry. The term "topology" was introduced by Johann Benedict Listing in 1847, although he had used the term in correspondence some years earlier instead of previously used "Analysis situs". The foundation of this science, for a space of any dimension, was created by Henri Poincaré. His first article on this topic appeared in 1894. In the 1930s, James Waddell Alexander II and Hassler Whitney first expressed the idea that a surface is a topological space that is locally like a Euclidean plane. Topological spaces were first defined by Felix Hausdorff in 1914 in his seminal "Principles of Set Theory". Metric spaces had been defined earlier in 1906 by Maurice Fréchet, though it was Hausdorff who popularised the term "metric space" ( de , metrischer Raum).

# Definitions

The utility of the concept of a ''topology'' is shown by the fact that there are several equivalent definitions of this mathematical structure. Thus one chooses the
axiomatization In mathematics and logic, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory is a consistent, relatively-self-contained body of knowledge which usually contai ...
suited for the application. The most commonly used is that in terms of , but perhaps more intuitive is that in terms of and so this is given first.

## Definition via neighbourhoods

This axiomatization is due to Felix Hausdorff. Let $X$ be a set; the elements of $X$ are usually called , though they can be any mathematical object. We allow $X$ to be empty. Let $\mathcal$ be a function assigning to each $x$ (point) in $X$ a non-empty collection $\mathcal\left(x\right)$ of subsets of $X.$ The elements of $\mathcal\left(x\right)$ will be called of $x$ with respect to $\mathcal$ (or, simply, ). The function $\mathcal$ is called a neighbourhood topology if the axioms below are satisfied; and then $X$ with $\mathcal$ is called a topological space. # If $N$ is a neighbourhood of $x$ (i.e., $N \in \mathcal\left(x\right)$), then $x \in N.$ In other words, each point belongs to every one of its neighbourhoods. # If $N$ is a subset of $X$ and includes a neighbourhood of $x,$ then $N$ is a neighbourhood of $x.$ I.e., every superset of a neighbourhood of a point $x \in X$ is again a neighbourhood of $x.$ # The intersection of two neighbourhoods of $x$ is a neighbourhood of $x.$ # Any neighbourhood $N$ of $x$ includes a neighbourhood $M$ of $x$ such that $N$ is a neighbourhood of each point of $M.$ The first three axioms for neighbourhoods have a clear meaning. The fourth axiom has a very important use in the structure of the theory, that of linking together the neighbourhoods of different points of $X.$ A standard example of such a system of neighbourhoods is for the real line $\R,$ where a subset $N$ of $\R$ is defined to be a of a real number $x$ if it includes an open interval containing $x.$ Given such a structure, a subset $U$ of $X$ is defined to be open if $U$ is a neighbourhood of all points in $U.$ The open sets then satisfy the axioms given below. Conversely, when given the open sets of a topological space, the neighbourhoods satisfying the above axioms can be recovered by defining $N$ to be a neighbourhood of $x$ if $N$ includes an open set $U$ such that $x \in U.$

## Definition via open sets

A ''topology'' on a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
may be defined as a collection $\tau$ of subsets of , called open sets and satisfying the following axioms: # The empty set and $X$ itself belong to $\tau.$ # Any arbitrary (finite or infinite) union of members of $\tau$ belongs to $\tau.$ # The intersection of any finite number of members of $\tau$ belongs to $\tau.$ As this definition of a topology is the most commonly used, the set $\tau$ of the open sets is commonly called a topology on $X.$ A subset $C \subseteq X$ is said to be in $\left(X, \tau\right)$ if its complement $X \setminus C$ is an open set.

### Examples of topologies

# Given $X = \,$ the
trivial Trivia is information and data that are considered to be of little value. It can be contrasted with general knowledge and common sense. Latin Etymology The ancient Romans used the word ''triviae'' to describe where one road split or fork ...
or topology on $X$ is the family $\tau = \ = \$ consisting of only the two subsets of $X$ required by the axioms forms a topology of $X.$ # Given $X = \,$ the family $\tau = \ = \$ of six subsets of $X$ forms another topology of $X.$ # Given $X = \,$ the discrete topology on $X$ is the power set of $X,$ which is the family $\tau = \wp\left(X\right)$ consisting of all possible subsets of $X.$ In this case the topological space $\left(X, \tau\right)$ is called a . # Given $X = \Z,$ the set of integers, the family $\tau$ of all finite subsets of the integers plus $\Z$ itself is a topology, because (for example) the union of all finite sets not containing zero is not finite but is also not all of $\Z,$ and so it cannot be in $\tau.$

## Definition via closed sets

Using de Morgan's laws, the above axioms defining open sets 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 finite number 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 $\tau$ of closed subsets of $X.$ Thus the sets in the topology $\tau$ are the closed sets, and their complements in $X$ are the open sets.

## Other definitions

There are many other equivalent ways to define a topological space: in other words the concepts of neighbourhood, or that of open or closed sets can be reconstructed from other starting points and satisfy the correct axioms. 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 $X.$ 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 $\tau_1$ is also in a topology $\tau_2$ and $\tau_1$ is a subset of $\tau_2,$ we say that $\tau_2$is than $\tau_1,$ and $\tau_1$ is than $\tau_2.$ A proof that 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 and are sometimes used in place of finer and coarser, respectively. The terms and 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 = \left\$ is a collection of topologies on $X,$ then the meet of $F$ is the intersection of $F,$ and the
join Join may refer to: * Join (law), to include additional counts or additional defendants on an indictment *In mathematics: ** Join (mathematics), a least upper bound of sets orders in lattice theory ** Join (topology), an operation combining two topo ...
of $F$ is the meet of the collection of all topologies on $X$ that contain every member of $F.$

# Continuous functions

A function $f : X \to Y$ between topological spaces is called
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
if for every $x \in X$ and every neighbourhood $N$ of $f\left(x\right)$ there is a neighbourhood $M$ of $x$ such that $f\left(M\right) \subseteq N.$ This relates easily to the usual definition in analysis. Equivalently, $f$ is continuous if the inverse image of every open set is open. This is an attempt to capture the intuition that there are no "jumps" or "separations" in the function. A homeomorphism is a bijection that is continuous and whose
inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when a ...
is also continuous. Two spaces are called if there exists a homeomorphism between them. From the standpoint of topology, homeomorphic spaces are essentially identical. In category theory, one of the fundamental categories is Top, which denotes the category of topological spaces whose
objects Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Object (abstract), an object which does not exist at any particular time or place ** Physical object, an identifiable collection of matter * Goal, an ai ...
are topological spaces and whose morphisms are continuous functions. The attempt to classify the objects of this category ( up to homeomorphism) by invariants has motivated areas of research, such as homotopy theory, homology theory, and K-theory.

# 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, often topological spaces must be
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the ma ...
s where limit points are unique.

## Metric spaces

Metric spaces embody a metric, a precise notion of distance between points. 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. On a finite-dimensional vector space this topology is the same for all norms. 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 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, the
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
s $\R^n$ can be given a topology. In the usual topology on $\R^n$ the basic open sets are the open balls. Similarly, $\C,$ the set of complex numbers, and $\C^n$ have a standard topology in which the basic open sets are open balls.

## Other spaces

If $\Gamma$ is a filter on a set $X$ then $\ \cup \Gamma$ is a topology on $X.$ Many sets of linear operators in
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defi ...
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, every simplex and every simplicial complex inherits a natural topology from . 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 set In mathematics, a solution set is the set of values that satisfy a given set of equations or inequalities. For example, for a set of polynomials over a ring , the solution set is the subset of on which the polynomials all vanish (evaluate t ...
s of systems of polynomial equations. A
linear graph In the mathematical field of graph theory, a path graph or linear graph is a graph whose vertices can be listed in the order such that the edges are where . Equivalently, a path with at least two vertices is connected and has two terminal ...
has a natural topology that generalizes many of the geometric aspects of graphs with vertices and edges. The Sierpiński space is the simplest 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 called
finite topological space In mathematics, a finite topological space is a topological space for which the underlying point set is finite. That is, it is a topological space which has only finitely many elements. Finite topological spaces are often used to provide example ...
s. Finite spaces are sometimes used to provide examples or counterexamples to conjectures about topological spaces in general. Any 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 T1 topology on any infinite set. Any set can be given the cocountable topology, in which a set is defined as open if it is either empty or its complement is countable. When the set is uncountable, this topology serves as a 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 $\left[a, b\right).$ 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 $\Gamma$ is an ordinal number, then the set $\Gamma = \left[0, \Gamma\right)$ may be endowed with the order topology generated by the intervals $\left(a, b\right),$ $\left[0, b\right),$ and $\left(a, \Gamma\right)$ where $a$ and $b$ are elements of $\Gamma.$
Outer space Outer space, commonly shortened to space, is the expanse that exists beyond Earth and its atmosphere and between celestial bodies. Outer space is not completely empty—it is a near-perfect vacuum containing a low density of particles, pred ...
of a free group $F_n$ consists of the so-called "marked metric graph structures" of volume 1 on $F_n.$

# 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 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. 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 $f : X \to Y$ 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. The Vietoris topology on the set of all non-empty subsets of a topological space $X,$ named for
Leopold Vietoris Leopold Vietoris (; ; 4 June 1891 – 9 April 2002) was an Austrian mathematician, World War I veteran and supercentenarian. He was born in Radkersburg and died in Innsbruck. He was known for his contributions to topology—notably the Mayer– ...
, is generated by the following basis: for every $n$-tuple $U_1, \ldots, U_n$ of open sets in $X,$ we construct a basis set consisting of all subsets of the union of the $U_i$ that have non-empty intersections with each $U_i.$ The Fell topology on the set of all non-empty closed subsets of a locally compact Polish space $X$ is a variant of the Vietoris topology, and is named after mathematician James Fell. It is generated by the following basis: for every $n$-tuple $U_1, \ldots, U_n$ of open sets in $X$ and for every compact set $K,$ the set of all subsets of $X$ that are disjoint from $K$ and have nonempty intersections with each $U_i$ is a member of the basis.

# 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 not shared by them. Examples of such properties include connectedness, compactness, and various
separation axiom 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. Some of these restrictions are given by the separation axioms. These are sometim ...
s. For algebraic invariants see algebraic topology.

# 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 that is not finite, we often have a natural topology 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 ring In mathematics, a topological ring is a ring R that is also a topological space such that both the addition and the multiplication are continuous as maps: R \times R \to R where R \times R carries the product topology. That means R is an additive ...
s and local fields.

# Topological spaces with order structure

* Spectral: A space is ''
spectral ''Spectral'' is a 2016 3D military science fiction, supernatural horror fantasy and action-adventure thriller war film directed by Nic Mathieu. Written by himself, Ian Fried, and George Nolfi from a story by Fried and Mathieu. The film stars J ...
'' if and only if it is the prime spectrum of a ring ( Hochster theorem). * Specialization preorder: In a space the ''specialization preorder'' (or ''canonical preorder'') is defined by $x \leq y$ if and only if $\operatorname\ \subseteq \operatorname\,$ where $\operatorname$ denotes an operator satisfying the Kuratowski closure axioms.

* *
Complete Heyting algebra In mathematics, especially in order theory, a complete Heyting algebra is a Heyting algebra that is complete as a lattice. Complete Heyting algebras are the objects of three different categories; the category CHey, the category Loc of locales, ...
– The system of all open sets of a given topological space ordered by inclusion is a complete Heyting algebra. * * * * * * * * * *

# Bibliography

* * Bredon, Glen E., ''Topology and Geometry'' (Graduate Texts in Mathematics), Springer; 1st edition (October 17, 1997). . *
Bourbaki, Nicolas Nicolas Bourbaki () is the collective pseudonym of a group of mathematicians, predominantly French alumni of the École normale supérieure - PSL (ENS). Founded in 1934–1935, the Bourbaki group originally intended to prepare a new textbook ...
; ''Elements of Mathematics: General Topology'', Addison-Wesley (1966). * (3rd edition of differently titled books) * Čech, Eduard; ''Point Sets'', Academic Press (1969). * Fulton, William, ''Algebraic Topology'', (Graduate Texts in Mathematics), Springer; 1st edition (September 5, 1997). . * * * Lipschutz, Seymour; ''Schaum's Outline of General Topology'', McGraw-Hill; 1st edition (June 1, 1968). . * Munkres, James; ''Topology'', Prentice Hall; 2nd edition (December 28, 1999). . * Runde, Volker; ''A Taste of Topology (Universitext)'', Springer; 1st edition (July 6, 2005). . * * Steen, Lynn A. and Seebach, J. Arthur Jr.; ''
Counterexamples in Topology ''Counterexamples in Topology'' (1970, 2nd ed. 1978) is a book on mathematics by topologists Lynn Steen and J. Arthur Seebach, Jr. In the process of working on problems like the metrization problem, topologists (including Steen and Seebach) h ...
'', Holt, Rinehart and Winston (1970). . * *