TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...
, a topological space is, roughly speaking, a in which ''closeness'' is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set of
point Point or points may refer to: Places * Point, LewisImage:Point Western Isles NASA World Wind.png, Satellite image of Point Point ( gd, An Rubha), also known as the Eye Peninsula, is a peninsula some 11 km long in the Outer Hebrides (or Western I ...
s, along with a set of
neighbourhood A neighbourhood (British English, Hiberno-English, Hibernian English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographicall ...
s for each point, satisfying a set of
axiom An axiom, postulate or assumption is a statement that is taken to be , to serve as a or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' or 'that which comm ...
s relating points and neighbourhoods. A topological space is the most general type of a
mathematical space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
that allows for the definition of
limits Limit or Limits may refer to: Arts and media * Limit (music) In music theory, limit or harmonic limit is a way of characterizing the harmony found in a piece or genre (music), genre of music, or the harmonies that can be made using a particular ...
, continuity, and
connectedness In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. Other spaces, such as
Euclidean space Euclidean space is the fundamental space of . Originally, it was the of , but in modern there are Euclidean spaces of any nonnegative integer , including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduce ...
s,
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
s and
manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

s, are topological spaces with extra
structures A structure is an arrangement and organization of interrelated elements in a material object or system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A sy ...
, properties or constraints. Although very general, topological spaces are a fundamental concept used in virtually every branch of modern mathematics. The branch of mathematics that studies topological spaces in their own right is called
point-set topology , a useful example in point-set topology. It is connected but not path-connected. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algeb ...
or
general topology , a useful example in point-set topology. It is connected but not path-connected. In mathematics, general topology is the branch of topology that deals with the basic Set theory, set-theoretic definitions and constructions used in topology. It is t ...
.

# History

Around
1735 Events January–March * January 2 Events Pre-1600 *AD 69 – The Roman legions in Germania Superior refuse to swear loyalty to Galba. They rebel and proclaim Vitellius as emperor. * 366 – The Alemanni cross the frozen ...
,
Leonhard Euler Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who made important and influential discoveries in many branches of mathematics, such as infinitesimal c ...

discovered the
formula In , a formula is a concise way of expressing information symbolically, as in a mathematical formula or a . The informal use of the term ''formula'' in science refers to the . The plural of ''formula'' can be either ''formulas'' (from the mos ...

$V - E + F = 2$ relating the number of vertices, edges and faces of a
convex polyhedron A convex polytope is a special case of a polytope In elementary geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest bran ...
, and hence of a
planar graph In graph theory In mathematics, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph t ...

. The study and generalization of this formula, specifically by
Cauchy Baron Augustin-Louis Cauchy (; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He was ...

and L'Huilier, is at the origin of
topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

. In
1827 Events January–March * January 5 – The first regatta A regatta is a series of boat races. The term comes from the Venetian language regata meaning "contest" and typically describes racing events of Rowing (sport), rowed or sail ...
,
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician This is a List of German mathematician A mathematician is someone who uses an extensive knowledge of ma ...

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 Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of ...
’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." " and
Jordan Jordan ( ar, الأردن; tr. ' ), officially the Hashemite Kingdom of Jordan,; tr. ') is a country in Western Asia Western Asia, also West Asia, is the westernmost subregion of Asia. It is entirely a part of the Greater Middle East. It in ...
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 Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group ...
in his "
Erlangen Program In mathematics, the Erlangen program is a method of characterizing geometries based on group theory The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"titl ...
" (1872): the geometry invariants of arbitrary continuous transformation, a kind of geometry. The term "topology" was introduced by
Johann Benedict Listing Johann Benedict Listing (25 July 1808 – 24 December 1882) was a German mathematician. J. B. Listing was born in Frankfurt Frankfurt, officially Frankfurt am Main (; Hessian: ''Frangford am Maa'', " Frank ford on the Main"), is the ...
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 Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French French (french: français(e), link=no) may refer to: * Something of, from, or related to France France (), officially the French Repu ...
. His first article on this topic appeared in
1894 Events January–March * January 4 Events Pre-1600 * 46 BC – Julius Caesar Gaius Julius Caesar (; 12 July 100 BC – 15 March 44 BC) was a Roman people, Roman general and statesman who played a critical role in Crisis of the ...
. In the 1930s,
James Waddell Alexander II James Waddell Alexander II (September 19, 1888 September 23, 1971) was a mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics ...
and
Hassler Whitney Hassler Whitney (March 23, 1907 – May 10, 1989) was an American mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quanti ...
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 Felix Hausdorff (November 8, 1868 – January 26, 1942) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as q ...
in 1914 in his seminal "Principles of Set Theory".
Metric spaces Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement Mathematics * Metric (mathematics), an abstraction of the notion of ''distance'' in a metric space * Metric tensor, in differential geometr ...
had been defined earlier in 1906 by
Maurice FréchetMaurice may refer to: People *Saint Maurice Saint Maurice (also Moritz, Morris, or Mauritius; ) was the leader of the legendary Roman Theban Legion in the 3rd century, and one of the favorite and most widely venerated saints of that group. He w ...
, though it was Hausdorff who introduced the term "metric space".

# Definitions

The utility of the notion of a topology is shown by the fact that there are several equivalent definitions of this structure. Thus one chooses the
axiomatization In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 Felix Hausdorff (November 8, 1868 – January 26, 1942) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as q ...
. 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 Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
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
axiom An axiom, postulate or assumption is a statement that is taken to be , to serve as a or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' or 'that which comm ...

s 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
of a neighbourhood of a point $x \in X$ is again a neighbourhood of $x.$ # The
intersection The line (purple) in two points (red). The disk (yellow) intersects the line in the line segment between the two red points. In mathematics, the intersection of two or more objects is another, usually "smaller" object. Intuitively, the inter ...
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 is an ordered pair $\left(X, \tau\right),$ where $X$ is a set and $\tau$ is a collection of
subset In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s of $X,$ satisfying the following
axiom An axiom, postulate or assumption is a statement that is taken to be , to serve as a or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' or 'that which comm ...

s: # The
empty set #REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, cha ...

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 $\tau.$ The elements of $\tau$ are called open sets and the collection $\tau$ is called a topology on $X.$ A subset $C \subseteq X$ is said to be in $\left(X, \tau\right)$ if and only if its
complement A complement is often something that completes something else, or at least adds to it in some useful way. Thus it may be: * Complement (linguistics), a word or phrase having a particular syntactic role ** Subject complement, a word or phrase addi ...
$X \setminus C$ is an element of $\tau.$

### Examples of topologies

# Given $X = \,$ the trivial or topology on $X$ is the
family In , family (from la, familia) is a of people related either by (by recognized birth) or (by marriage or other relationship). The purpose of families is to maintain the well-being of its members and of society. Ideally, families would off ...
$\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 In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric objec ...
on $X$ is the
power set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
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 In propositional calculus, propositional logic and Boolean algebra, De Morgan's laws are a pair of transformation rules that are both Validity (logic), valid rule of inference, rules of inference. They are named after Augustus De Morgan, a 19th ...
, the above axioms defining open sets become axioms defining
closed set In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...
s: # 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 Kazimierz Kuratowski (; 2 February 1896 – 18 June 1980) was a Polish mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as ...
, which define the closed sets as the fixed points of an operator on the
power set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
of $X.$ A
net Net or net may refer to: Mathematics and physics * Net (mathematics) In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence. In essence, ...
is a generalisation of the concept of
sequence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

. 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
: if $F = \left\$ 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 $X$ that contain every member of $F.$

# Continuous functions

A
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
$f : X \to Y$ 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 ga ...
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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

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 and a donut (torus In geometry, a torus (plural tori, colloquially donut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanarity, coplanar with the circle. If the axis of ...
is a
bijection In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

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 add ...
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 Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
, one of the fundamental
categories Category, plural categories, may refer to: Philosophy and general uses *Categorization Categorization is the human ability and activity of recognizing shared features or similarities between the elements of the experience of the world (such ...
is Top, which denotes the
category of topological spaces In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
whose objects are topological spaces and whose
morphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

s are continuous functions. The attempt to classify the objects of this category (
up to Two mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
homeomorphism and a donut (torus In geometry, a torus (plural tori, colloquially donut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanarity, coplanar with the circle. If the axis of ...
) by invariants has motivated areas of research, such as
homotopy theoryIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

,
homology theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, and
K-theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
.

# 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 topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), an ...
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 topologyIn topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object ...
(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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), ...

s where limit points are unique.

## Metric spaces

Metric spaces embody a
metric METRIC (Mapping EvapoTranspiration at high Resolution with Internalized Calibration) is a computer model Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of or th ...
, a precise notion of distance between points. Every
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
. On a finite-dimensional
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
this topology is the same for all norms. There are many ways of defining a topology on $\R,$ the set of
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s. The standard topology on $\R$ is generated by the . The set of all open intervals forms a
base Base or BASE may refer to: Brands and enterprises * Base (mobile telephony provider), a Belgian mobile telecommunications operator *Base CRM Base CRM (originally Future Simple or PipeJump) is an enterprise software company based in Mountain Vie ...
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 . Originally, it was the of , but in modern there are Euclidean spaces of any nonnegative integer , including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduce ...
s $\R^n$ can be given a topology. In the usual topology on $\R^n$ the basic open sets are the open
ball A ball is a round object (usually spherical of a sphere A sphere (from Greek language, Greek —, "globe, ball") is a geometrical object in three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional s ...
s. Similarly, $\C,$ the set of
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

s, and $\C^n$ have a standard topology in which the basic open sets are open balls.

## Proximity spaces

Proximity space Proximity may refer to: * Distance Distance is a numerical measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be used ...
s provide a notion of closeness of two sets.

## Uniform spaces

Uniform spaces axiomatize ordering the distance between distinct points.

## Function spaces

A topological space in which the are functions is called a
function space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
.

## Cauchy spaces

Cauchy space In general topology and mathematical analysis, analysis, a Cauchy space is a generalization of metric spaces and uniform spaces for which the notion of Cauchy convergence still makes sense. Cauchy spaces were introduced by H. H. Keller in 1968, as ...
s axiomatize the ability to test whether a net is
Cauchy Baron Augustin-Louis Cauchy (; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He was ...
. Cauchy spaces provide a general setting for studying completions.

## Convergence spaces

Convergence space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s capture some of the features of convergence of
filters Filter, filtering or filters may refer to: Science and technology Device * Filter (chemistry), a device which separates solids from fluids (liquids or gases) by adding a medium through which only the fluid can pass ** Filter (aquarium), critical ...
.

## Grothendieck sites

Grothendieck site In category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), sp ...
s are
categories Category, plural categories, may refer to: Philosophy and general uses *Categorization Categorization is the human ability and activity of recognizing shared features or similarities between the elements of the experience of the world (such ...
with additional data axiomatizing whether a family of arrows covers an object. Sites are a general setting for defining sheaves.

## Other spaces

If $\Gamma$ is a
filter Filter, filtering or filters may refer to: Science and technology Device * Filter (chemistry), a device which separates solids from fluids (liquids or gases) by adding a medium through which only the fluid can pass ** Filter (aquarium), critical ...
on a set $X$ then $\ \cup \Gamma$ is a topology on $X.$ Many sets of
linear operator In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
s in
functional analysis Functional analysis is a branch of mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), ...
are endowed with topologies that are defined by specifying when a particular sequence of functions converges to the zero function. Any
local field In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
has a topology native to it, and this can be extended to vector spaces over that field. Every
manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

has a
natural topology In any domain of mathematics, a space has a natural topology if there is a topology on the space which is "best adapted" to its study within the domain in question. In many cases this imprecise definition means little more than the assertion that t ...
since it is locally Euclidean. Similarly, every
simplex In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...

and every
simplicial complex In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
inherits a natural topology from . The
Zariski topology In algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zero of a function, zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commut ...
is defined algebraically on the
spectrum of a ring In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
or an
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematic ...
. On $\R^n$ or $\C^n,$ the closed sets of the Zariski topology are the
solution set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s of systems of
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

equations. A linear graph has a natural topology that generalizes many of the geometric aspects of
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...
s with vertices and . The Sierpiński space is the simplest non-discrete topological space. It has important relations to the
theory of computation In theoretical computer science Theoretical computer science (TCS) is a subset of general computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation a ...
and semantics. There exist numerous topologies on any given
finite set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
. Such spaces are called
finite topological space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
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 In mathematics, the lower limit topology or right half-open interval topology is a topological space, topology defined on the set \mathbb of real numbers; it is different from the standard topology on \mathbb (generated by the open intervals) and h ...
. 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 Earth is the third planet from the Sun and the only astronomical object known to harbour and support life. 29.2% of Earth's surface is land consisting ...
of a
free group for the free group on two generators would look like. Each vertex represents an element of the free group, and each edge represents multiplication by ''a'' or ''b''. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the stud ...
$F_n$ 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 topologyIn topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a Topological_space#Definitions, topology induced from that of ''X'' called the subspace topology (or the relative ...

in which the open sets are the intersections of the open sets of the larger space with the subset. For any
indexed family In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
of topological spaces, the product can be given the
product topology Product may refer to: Business * Product (business) In marketing, a product is an object or system made available for consumer use; it is anything that can be offered to a Market (economics), market to satisfy the desire or need of a customer ...
, 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
, then the quotient topology on $Y$ is the collection of subsets of $Y$ that have open
inverse image In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

s under $f.$ In other words, the
quotient topology In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), ...
is the finest topology on $Y$ for which $f$ is continuous. A common example of a quotient topology is when an
equivalence relation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
is defined on the topological space $X.$ The map $f$ is then the natural projection onto the set of
equivalence class In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
es. 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 Austria Austria (, ; german: Österreich ), officially the Republic of Austria (german: Republik Österreich, links=no, ), is a landlocked Eastern Alps, East Alpine country in ...
, 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
Polish space In the mathematical discipline of general topology, a Polish space is a separable space, separable Completely metrizable space, completely metrizable topological space; that is, a space homeomorphic to a Complete space, complete metric space that ha ...
$X$ 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 Two mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
homeomorphism, by their
topological propertiesIn topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object ...
. 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
,
compactness In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, and various separation axioms. For algebraic invariants see algebraic topology.

# Topological spaces with algebraic structure

For any Algebraic structure, 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 rings and
local field In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
s.

# Topological spaces with order structure

* Spectral. A space is Spectral space, spectral if and only if it is the prime
spectrum of a ring In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
(Melvin Hochster, Hochster theorem). * Specialization preorder. In a space the Specialization preorder, specialization (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 Kazimierz Kuratowski (; 2 February 1896 – 18 June 1980) was a Polish mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as ...
.