topological spaces

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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 Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
. More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of
neighbourhood A neighbourhood (British English British English (BrE, en-GB, or BE) is, according to Oxford Dictionaries, " English as used in Great Britain, as distinct from that used elsewhere". More narrowly, it can refer specifically to the En ...
s for each point that satisfy some
axiom An axiom, postulate, or assumption is a statement (logic), statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that whi ...
s formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through
open set 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 ...
s, 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 Limit or Limits may refer to: Arts and media * Limit (manga), ''Limit'' (manga), a manga by Keiko Suenobu * Limit (film), ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * Limit (song), "Limit" (song), a 201 ...
, 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, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
s,
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functio ...
s and
manifold 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 i ...
s. 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 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 the foundation of most other branches of topology, including differential topolog ...
.

# History

Around
1735 Events January–March * January 2 Events Pre-1600 *AD 69, 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 ...
,
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
discovered the
formula In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the terminology, term ''formula'' in science refers to the Commensurability (philosophy o ...
$V - E + F = 2$ 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, a polytope is a geometric object with Flat (geometry), flat sides (''Face (geometry), faces''). Polytopes are the generalization of three-dimensional polyhedron, polyhe ...
, and hence of a
planar graph In graph theory, a planar graph is a graph (discrete mathematics), graph that can be graph embedding, embedded in the plane (geometry), plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. I ...
. The study and generalization of this formula, specifically by Cauchy (1789-1857) and L'Huilier (1750-1840), boosted the study of
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
. In
1827 Events January–March * January 5 – The first regatta in Australia is held, taking place on Tasmania (called at the time ''Van Diemen's Land''), on the River Derwent (Tasmania), River Derwent at Hobart. * January 15 – Furman Univers ...
,
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
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 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 geometry, geometries based on group theory and projective geometry. It was published by Felix Klein in 1872 as ''Vergleichende Betrachtungen über neuere geometrische Forschungen.'' ...
" (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 and died in Göttingen. He first introduced the term "topology" to replace the older term "geometria situs" (also called ...
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 Events January–March * January 4 – Franco-Russian Alliance, A military alliance is established between the French Third Republic and the Russian Empire. * January 7 – William Kennedy Dickson receives a patent for motion pictu ...
. In the 1930s,
James Waddell Alexander II James Waddell Alexander II (September 19, 1888 September 23, 1971) was a mathematician and topologist of the pre-World War II era and part of an influential Princeton topology elite, which included Oswald Veblen, Solomon Lefschetz, and others. ...
and
Hassler Whitney Hassler Whitney (March 23, 1907 – May 10, 1989) was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersion (mathematics), immersions, characteristic classes, and ...
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 in their work, typically to solve mathematical problems. Mathematicians are c ...
in 1914 in his seminal "Principles of Set Theory".
Metric spaces In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functio ...
had been defined earlier in 1906 by
Maurice Fréchet Maurice may refer to: People *Saint Maurice Saint Maurice (also Moritz, Morris, or Mauritius; ) was an Egyptians, Egyptian military leader who headed the legendary Theban Legion of Roman Empire, Rome in the 3rd century, and is one of the favo ...
, 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 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 in their work, typically to solve mathematical problems. Mathematicians are c ...
. 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
axiom An axiom, postulate, or assumption is a statement (logic), statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that whi ...
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, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
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 may be defined as a collection $\tau$ of
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
s of , called open sets and satisfying the following axioms: # The
empty set 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 ...
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 A complement is something that completes something else. Complement may refer specifically to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Complement (music)#Aggregate complementation, Aggregate c ...
$X \setminus C$ is an open set.

### Examples of topologies

# Given $X = \,$ the trivial or topology on $X$ is the
family Family (from la, familia) is a Social group, group of people related either by consanguinity (by recognized birth) or Affinity (law), affinity (by marriage or other relationship). The purpose of the family is to maintain the well-being of its ...
$\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, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are ''Isolated point, isolated'' from each other in a certain sense. The discrete topology is ...
on $X$ is the
power set 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 ...
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, also known as De Morgan's theorem, are a pair of transformation rules that are both Validity (logic), valid rule of inference, rules of inference. They are na ...
, the above axioms defining open sets become axioms defining
closed set In geometry, topology, and related branches of mathematics, a closed set is a Set (mathematics), set whose complement (set theory), complement is an open set. In a topological space, a closed set can be defined as a set which contains all its lim ...
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, which define the closed sets as the fixed points of an operator on the
power set 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 ...
of $X.$ A
net Net or net may refer to: Mathematics and physics * Net (mathematics), a filter-like topological generalization of a sequence * Net, a linear system of divisors of dimension 2 * Net (polyhedron), an arrangement of polygons that can be folded up ...
is a generalisation of the concept of
sequence In mathematics, a sequence is an enumerated collection of mathematical object, objects in which repetitions are allowed and order theory, order matters. Like a Set (mathematics), set, it contains Element (mathematics), members (also called ''eleme ...
. 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, a complete lattice is a partially ordered set in which ''all'' subsets have both a supremum (join) and an infimum (meet). A lattice which satisfies at least one of these properties is known as a ''conditionally complete lattice.'' S ...
: 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 $f : X \to Y$ between topological spaces is called continuous 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 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 ...
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 In the mathematics, mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and Continuous function#Continuous functions between topological spaces, continuous function between topologic ...
is a
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function (mathematics), function between the elements of two set (mathematics), sets, where each element of one set is pair ...
that is continuous and whose inverse 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 is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, categ ...
, one of the fundamental categories is Top, which denotes the category of topological spaces whose objects are topological spaces and whose
morphism 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 mod ...
s are continuous functions. The attempt to classify the objects of this category (
up to Two Mathematical object, mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' wi ...
homeomorphism In the mathematics, mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and Continuous function#Continuous functions between topological spaces, continuous function between topologic ...
) by invariants has motivated areas of research, such as
homotopy theory In mathematics, homotopy theory is a systematic study of situations in which Map (mathematics), maps can come with homotopy, homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent disciplin ...
,
homology theory In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or module (mathematics), modules, with other mathematical objects such as topological spaces. Homology groups were originally define ...
, 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 topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are ''Isolated point, isolated'' from each other in a certain sense. The discrete topology is ...
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 neighbourhood (mathematics), neighbourhoods of each which are disjoint s ...
s where limit points are unique.

## Metric spaces

Metric spaces embody a
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathema ...
, a precise notion of distance between points. Every
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functio ...
can be given a metric topology, in which the basic open sets are open balls defined by the metric. This is the standard topology on any
normed vector space In mathematics, a normed vector space or normed space is a vector space over the Real number, real or Complex number, complex numbers, on which a Norm (mathematics), norm is defined. A norm is the formalization and the generalization to real ve ...
. On a finite-dimensional
vector space 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 ...
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, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
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, the
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
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 sphere, spherical, but can sometimes be ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be ...
s. Similarly, $\C,$ the set of
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
s, 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 operator 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 ...
s in
functional analysis Functional analysis is a branch of mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, ...
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 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 i ...
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 ...
since it is locally Euclidean. Similarly, every
simplex In geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of ...
and every
simplicial complex In mathematics, a simplicial complex is a Set (mathematics), set composed of Point (geometry), points, line segments, triangles, and their Simplex, ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with ...
inherits a natural topology from . The
Zariski topology In algebraic geometry and commutative algebra, the Zariski topology is a topology (structure), topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the Real analysis, real or comp ...
is defined algebraically on the
spectrum of a ring In commutative algebra, the prime spectrum (or simply the spectrum) of a Ring (mathematics), ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological spac ...
or an
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the solution set, set of solutions of a system of polynomial equations over the real number, r ...
. On $\R^n$ or $\C^n,$ the closed sets of the Zariski topology are the solution sets of systems of
polynomial In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtrac ...
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 discre ...
s with vertices and edges. The Sierpiński space is the simplest non-discrete topological space. It has important relations to the
theory of computation In theoretical computer science and mathematics, the theory of computation is the branch that deals with what problems can be solved on a model of computation, using an algorithm, how algorithmic efficiency, efficiently they can be solved or t ...
and semantics. There exist numerous topologies on any given
finite set 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 mo ...
. Such spaces are called finite topological spaces. 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 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 mod ...
. 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 Earth atmosphere, its atmosphere and between astronomical object, celestial bodies. Outer space is not completely empty—it is a Ultra-high vacuum, near-per ...
of a
free group In mathematics, the free group ''F'S'' over a given set ''S'' consists of all Word (group theory), words that can be built from members of ''S'', considering two words to be different unless their equality follows from the Group (mathematics) ...
$F_n$ 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 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 relat ...
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 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 m ...
of topological spaces, the product can be given the
product topology In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemin ...
, 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 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 m ...
function, then the quotient topology on $Y$ is the collection of subsets of $Y$ that have open
inverse image 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 ...
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 an
equivalence relation 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 i ...
is defined on the topological space $X.$ The map $f$ is then the natural projection onto the set of
equivalence class 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 ...
es. 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 $U_1, \ldots, U_n$ 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 topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
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 Two Mathematical object, mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' wi ...
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 In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed set, closed and bounded set, bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or ...
, and various separation axioms. For algebraic invariants see
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
.

# 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 group 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 ...
s,
topological vector space 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 mo ...
s, topological rings and local fields.

# Topological spaces with order structure

* Spectral: A space is '' spectral'' if and only if it is the prime
spectrum of a ring In commutative algebra, the prime spectrum (or simply the spectrum) of a Ring (mathematics), ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological spac ...
( 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 completeness (order theory), complete as a lattice (order), lattice. Complete Heyting algebras are the object (category theory), objects of three d ...
– 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; ''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'', Holt, Rinehart and Winston (1970). . * *