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
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
neighbourhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural a ...
s 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 set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
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,
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 space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
s and
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
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
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geomet ...
or
general topology.
History
Around
1735,
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 ...
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 term ''formula'' in science refers to the general construct of a relationship betwe ...
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 mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
. 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
be a set; the elements of
are usually called , though they can be any mathematical object. We allow
to be empty. Let
be a
function assigning to each
(point) in
a non-empty collection
of subsets of
The elements of
will be called of
with respect to
(or, simply, ). The function
is called a
neighbourhood topology if the
axioms below are satisfied; and then
with
is called a topological space.
# If
is a neighbourhood of
(i.e.,
), then
In other words, each point belongs to every one of its neighbourhoods.
# If
is a subset of
and includes a neighbourhood of
then
is a neighbourhood of
I.e., every
superset of a neighbourhood of a point
is again a neighbourhood of
# The
intersection of two neighbourhoods of
is a neighbourhood of
# Any neighbourhood
of
includes a neighbourhood
of
such that
is a neighbourhood of each point of
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
A standard example of such a system of neighbourhoods is for the real line
where a subset
of
is defined to be a of a real number
if it includes an open interval containing
Given such a structure, a subset
of
is defined to be open if
is a neighbourhood of all points in
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
to be a neighbourhood of
if
includes an open set
such that
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
of
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all 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 unequal, then ''A'' is a proper subset of ...
s of , called open sets and satisfying the following axioms:
# The
empty set and
itself belong to
# Any arbitrary (finite or infinite)
union of members of
belongs to
# The intersection of any finite number of members of
belongs to
As this definition of a topology is the most commonly used, the set
of the open sets is commonly called a topology on
A subset
is said to be in
if its
complement is an open set.
Examples of topologies
# Given
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
is the
family
Family (from la, familia) is a group of people related either by consanguinity (by recognized birth) or affinity (by marriage or other relationship). The purpose of the family is to maintain the well-being of its members and of society. Idea ...
consisting of only the two subsets of
required by the axioms forms a topology of
# Given
the family
of six subsets of
forms another topology of
# Given
the
discrete topology on
is the
power set of
which is the family
consisting of all possible subsets of
In this case the topological space
is called a .
# Given
the set of integers, the family
of all finite subsets of the integers plus
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
and so it cannot be in
Definition via closed sets
Using
de Morgan's laws, 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 whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a ...
s:
# The empty set and
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
together with a collection
of closed subsets of
Thus the sets in the topology
are the closed sets, and their complements in
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
A
net is a generalisation of the concept of
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
. A topology is completely determined if for every net in
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
is also in a topology
and
is a subset of
we say that
is than
and
is than
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
forms a
complete lattice: if
is a collection of topologies on
then the
meet of
is the intersection of
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
is the meet of the collection of all topologies on
that contain every member of
Continuous functions
A
function 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
and every neighbourhood
of
there is a neighbourhood
of
such that
This relates easily to the usual definition in analysis. Equivalently,
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
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isom ...
is a
bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
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
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, ca ...
, 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
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isom ...
) 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
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 mathe ...
, a precise notion of distance between points.
Every
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
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
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
this topology is the same for all norms.
There are many ways of defining a topology on
the set of
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s. The standard topology on
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
can be given a topology. In the usual topology on
the basic open sets are the open
balls. Similarly,
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 fo ...
s, and
have a standard topology in which the basic open sets are open balls.
Proximity spaces
Uniform spaces
Function spaces
Cauchy spaces
Convergence spaces
Grothendieck sites
Other spaces
If
is a
filter on a set
then
is a topology on
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
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
has a
natural topology since it is locally Euclidean. Similarly, every
simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
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
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
. On
or
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
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
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
This topology on
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
is an ordinal number, then the set
may be endowed with the order topology generated by the intervals
and
where
and
are elements of
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 consists of the so-called "marked metric graph structures" of volume 1 on
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
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-seem ...
, 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
is a topological space and
is a set, and if
is a
surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element o ...
function, then the quotient topology on
is the collection of subsets of
that have open
inverse images under
In other words, the
quotient topology is the finest topology on
for which
is continuous. A common example of a quotient topology is when an
equivalence relation is defined on the topological space
The map
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
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
-tuple
of open sets in
we construct a basis set consisting of all subsets of the union of the
that have non-empty intersections with each
The Fell topology on the set of all non-empty closed subsets of a
locally compact Polish space is a variant of the Vietoris topology, and is named after mathematician James Fell. It is generated by the following basis: for every
-tuple
of open sets in
and for every compact set
the set of all subsets of
that are disjoint from
and have nonempty intersections with each
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
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
.
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, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
s,
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
if and only if
where
denotes an operator satisfying the
Kuratowski closure axioms.
See also
*
*
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.
*
*
*
*
*
*
*
*
*
*
Citations
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). .
*
*
External links
*
{{Authority control
General topology