Krull topology

In mathematics, pro-finite groups are groups that are in a certain sense assembled from finite groups; they share many properties with the finite groups.

Definition

Formally, a pro-finite group is a group that is isomorphic to the inverse limit of an inverse system of finite groups. Pro-finite groups are naturally regarded as topological groups: each of the finite groups carries the discrete topology, and since the inverse limit is a subset of the product of these discrete spaces, it inherits a topology which turns it into a topological group. Equivalently, one can define pro-finite groups to be the topological groups that are Hausdorff, compact and totally disconnected.

Examples

- Finite groups are pro-finite, if given the discrete topology.
- The group of p-adic integers Z_{p under addition is pro-finite. It is the inverse limit of the finite groups Z/pⁿZ where n ranges over all natural numbers and the natural maps Z/pⁿZ → Z/p^mZ (n≥m) are used for the limit process. The topology on this pro-finite group is the same as the topology arising from the p-adic valuation on Z_{p.

- The Galois theory of field extensions of infinite degree gives rise naturally to Galois groups that are pro-finite. Specifically, if L/K is a Galois extension, we consider the group G = Gal(L/K) consisting of all field automorphisms of L which keep all elements of K fixed. This group is the inverse limit of the finite groups Gal(F/K), where F ranges over all intermediate fields such that F/K is a finite Galois extension. For the limit process, we use the restriction homomorphisms Gal(F₁/K) → Gal(F₂/K), where F₂ ⊆ F₁. The topology we obtain on Gal(L/K) is known as the Krull topology after Wolfgang Krull. Interestingly, Waterhouse showed that every pro-finite group is isomorphic to one arising from the Galois theory of some field K, but one cannot control which field K will be in this case. In fact, given a fixed field K, one does not know in general if even the finite groups occur as Galois groups over K. This is the Inverse Galois Problem for a field K.

- The fundamental groups considered in algebraic geometry are also pro-finite groups, roughly speaking because the algebra can only 'see' finite coverings of an algebraic variety. The fundamental groups of algebraic topology, however, are in general not pro-finite.

Properties and facts

Every product of (arbitrarily many) pro-finite groups is pro-finite; the topology arising from the pro-finiteness agrees with the product topology.
Every closed subgroup of a pro-finite group is itself pro-finite; the topology arising from the pro-finiteness agrees with the subspace topology. If N is a closed normal subgroup of a pro-finite group G, then the factor group G/N is pro-finite; the topology arising from the pro-finiteness agrees with the quotient topology.

Since every pro-finite group G is compact Hausdorff, we have a Haar measure on G, which allows us to measure the "size" of subsets of G, compute certain probabilities, and integrate functions on G.

Pro-finite completion

Given an arbitrary group G, there is a related pro-finite group G^{^}, the pro-finite completion of G. It is defined as the inverse limit of the groups G/N, where N runs through the normal subgroups in G of finite index (these normal subgroups are partially ordered by inclusion, which translates into an inverse system of natural homomorphisms between them). There is a natural homomorphism η : G → G^{^}, and the image of G under this homomorphism is dense in G^{^}. The homomorphism η is injective if and only if the group G is residually finite (i.e. iff for every non-identity element g in G there exists a normal subgroup N in G of finite index that doesn't contain g). The homomorphism η is characterized by the following universal property: given any pro-finite group H and any group homomorphism f : G → H, there exists a unique continuous group homomorphism g : G^{^} → H with f = gη.

Ind-finite groups

There is a notion of ind-finite group, which is the concept dual to pro-finite groups; i.e. a group G is ind-finite if it is the direct limit of finite groups. The usual terminology is different: a group G is called locally finite if every finitely-generated subgroup is finite. This is equivalent, in fact, to being 'ind-finite'.

By applying Pontryagin duality, one can see that abelian pro-finite groups are in duality with locally finite discrete abelian groups. The latter are just the abelian torsion groups.

See also: locally cyclic group.

Further reading

- Hendrik Lenstra: Profinite Groups, talk given in Oberwolfach, November 2003. [http://math.berkeley.edu/~jvoight/notes/oberwolfach/Lenstra-Profinite.pdf online version].

- Alexander Lubotzky: review of several books about pro-finite groups. Bulletin of the American Mathematical Society, 38 (2001), pages 475-479. [http://www.ams.org/bull/2001-38-04/S0273-0979-01-00914-4/home.html online version].

- J. P. Serre, Cohomologie Galoisienne. Springer Lecture Notes in Mathematics, vol. 5.

- William C. Waterhouse. Profinite groups are Galois groups. Proc. Amer. Math. Soc. 42 (1973), pp. 639–640.

Category:Group theory
Category:Topological groups

Mathematics

Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions.

Practical mathematics, in nearly every society, is used for such purposes as accounting, measuring land, or predicting astronomical events. Mathematical discovery or research often involves discovering and cataloging patterns, without regard for application. The remarkable fact that the "purest" mathematics often turns out to have practical applications is what Eugene Wigner has called "the unreasonable effectiveness of mathematics." Today, the natural sciences, engineering, economics, and medicine depend heavily on new mathematical discoveries.

The word "mathematics" comes from the Greek μάθημα (máthema) meaning "science, knowledge, or learning" and μαθηματικός (mathematikós) meaning "fond of learning". It is often abbreviated maths in Commonwealth English and math in North American English.

History
:Main article: History of mathematics

The evolution of mathematics might be seen to be an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction was probably that of numbers. The realization that two apples and two oranges do have something in common, namely that they fill the hands of exactly one person, was a breakthrough in human thought.
In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. Arithmetic (e.g. addition, subtraction, multiplication and division), naturally followed. Monolithic monuments testify to a knowledge of geometry.

Further steps need writing or some other system for recording numbers such as tallies or the knotted strings called khipu used by the Inca empire to store numerical data. Numeral systems have been many and diverse.

Historically, the major disciplines within mathematics arose, from the start of recorded history, out of the need to do calculations on taxation and commerce, to understand the relationships among numbers, to measure land, and to predict astronomical events. These needs can be roughly related to the broad subdivision of mathematics, into the studies of quantity, structure, space, and change.

Mathematics since has been much extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both.

Mathematical discoveries have been made throughout history and continue to be made today.

Inspiration, pure and applied mathematics, and aesthetics

Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in commerce, land measurement and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. Newton invented infinitesimal calculus and Feynman his Feynman path integral using a combination of reasoning and physical insight, and today's string theory also inspires new mathematics. Some mathematics is only relevant in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts.

As in most areas of study, the explosion of knowledge in the scientific age has led to specialization in mathematics. One major distinction is between pure mathematics and applied mathematics. Within applied mathematics, two major areas have split off and become disciplines in their own right, statistics and computer science.

Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty also in a clever proof, such as Euclid's proof that there are infinitely many prime numbers, and in a numerical method that speeds calculation, such as the fast Fourier transform. G. H. Hardy in "A Mathematicians Apology" expressed the belief that these esthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. Main article: Mathematical beauty.

Notation, language, and rigor

Mathematical writing is not easily accessible to the layperson. A Brief History of Time, Stephen Hawking's 1988 bestseller, contained a single mathematical equation. This was the author's compromise with the publisher's advice, that each equation would halve the sales.

The reasons for the inaccessibility even of carefully-expressed mathematics can be partially explained. Contemporary mathematicians strive to be as clear as possible in the things they say and especially in the things they write (this they have in common with lawyers). They refer to rigor. To accomplish rigor, mathematicians have extended natural language. There is precisely-defined vocabulary for referring to mathematical objects, and stating certain common relations. There is an accompanying mathematical notation which, like musical notation, has a definite content and also has a strict grammar (under the influence of computer science, more often now called syntax). Some of the terms used in mathematics are also common outside mathematics, such as ring, group and category; but are not such that one can infer the meanings. Some are specific to mathematics, such as homotopy and Hilbert space. It was said that Henri Poincaré was only elected to the Académie Française so that he could tell them how to define automorphe in their dictionary.

Rigor is fundamentally a matter of mathematical proof. Mathematicians want their theorems to follow mechanically from axioms by means of formal axiomatic reasoning. This is to avoid mistaken 'theorems', based on fallible intuitions; of which plenty of examples have occurred in the history of the subject (for example, in mathematical analysis).

Axioms in traditional thought were 'self-evident truths', but that conception turns out not to be workable in pushing the mathematical boundaries. At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (strong enough) axiom system has undecidable formulas; and so a final axiomatization of mathematics is unavailable. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.

Is mathematics a science?

Carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. The mathematician-physicist Leon M. Lederman has quipped: "The physicists defer only to mathematicians, and the mathematicians defer only to God (though you may be hard pressed to find a mathematician that modest)."

If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is public knowledge and thus includes mathematics. [http://info.med.yale.edu/therarad/summers/ziman.htm]

In any case, mathematics shares much in common with many fields in the physical sciences, notably
the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences.

Overview of fields of mathematics
As noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict astronomical events. These four needs can be roughly related to the broad subdivision of mathematics into the study of quantity, structure, space, and change (i.e. arithmetic, algebra, geometry and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations) and to the empirical mathematics of the various sciences (applied mathematics).

The study of quantity starts with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are characterized in arithmetic. The deeper properties of whole numbers are studied in number theory.

The study of structure began with investigations of Pythagorean triples. Neolithic monuments on the British Isles are constructed using Pythagorean triples. Eventually, this led to the invention of more abstract numbers, such as the square root of two. The deeper structural properties of numbers are studied in abstract algebra and the investigation of groups, rings, fields and other abstract number systems. Included is the important concept of vectors, generalized to vector spaces and studied in linear algebra. The study of vectors combines three of the fundamental areas of mathematics, quantity, structure, and space.

The study of space originates with geometry, beginning with Euclidean geometry. Trigonometry combines space and number. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Within differential geometry are the concepts of fiber bundles, calculus on manifolds. Within algebraic geometry is the description of geometric objects as solution sets of polynomal equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Lie groups are used to study space, structure, and change. Topology in all its many ramifications may be the greatest growth area in 20th century mathematics.

Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a most useful tool. The central concept used to describe a changing quantity is that of a function. Many problems lead quite naturally to relations between a quantity and its rate of change, and the methods of differential equations. The numbers used to represent continuous quantities are the real numbers, and the detailed study of their properties and the properties of real-valued functions is known as real analysis. These have been generalized, with the inclusion of the square root of negative one, to the complex numbers, which are studied in complex analysis. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior.

Beyond quantity, structure, space, and change are areas of pure mathematics that can be approached only by deductive reasoning. In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed. Mathematical logic, which divides into recursion theory, model theory, and proof theory, is now closely linked to computer science. When electronic computers were first conceived, several essential theoretical concepts in computer science were shaped by mathematicians, leading to the fields of computability theory, computational complexity theory, and information theory. Many of those topics are now investigated in theoretical computer science. Discrete mathematics is the common name for the fields of mathematics most generally useful in computer science.

An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis, and prediction of phenomena where chance plays a part. It is used in all the sciences. Numerical analysis investigates methods for using computers to efficiently solve a broad range of mathematical problems that are typically beyond human capacity, and taking rounding errors or other sources of error into account to obtain credible answers.

Major themes in mathematics
An alphabetical and subclassified list of mathematical topics is available. The following list of themes and links gives just one possible view. For a fuller treatment, see Areas of mathematics or the list of lists of mathematical topics.

Quantity
This starts from explicit measurements of sizes of numbers or sets, or ways to find such measurements.

:

:Number – Natural number – Integers – Rational numbers – Real numbers – Complex numbers – Hypercomplex numbers – Quaternions – Octonions – Sedenions – Hyperreal numbers – Surreal numbers – Ordinal numbers – Cardinal numbers – p-adic numbers – Integer sequences – Mathematical constants – Number names – Infinity – Base

Structure

:Pinning down ideas of size, symmetry, and mathematical structure.
:

:Abstract algebra – Number theory – Algebraic geometry – Group theory – Monoids – Analysis – Topology – Linear algebra – Graph theory – Universal algebra – Category theory – Order theory – Measure theory

Space

:A more visual approach to mathematics.
:

:Topology – Geometry – Trigonometry – Algebraic geometry – Differential geometry – Differential topology – Algebraic topology – Linear algebra – Fractal geometry

Change
:Ways to express and handle change in mathematical functions, and changes between numbers.
:

:Arithmetic – Calculus – Vector calculus – Analysis – Differential equations – Dynamical systems – Chaos theory – List of functions

Foundations and methods

:Approaches to understanding the nature of mathematics.

:philosophy of mathematics – mathematical intuitionism – mathematical constructivism – foundations of mathematics – set theory – symbolic logic – model theory – category theory – Logic – reverse mathematics – table of mathematical symbols

Discrete mathematics

:Discrete mathematics involves techniques that apply to objects that can only take on specific, separated values.

:

:Combinatorics – Naive set theory – Theory of computation– Cryptography – Graph theory

Applied mathematics

:Applied mathematics uses the full knowledge of mathematics to solve real-world problems.

:Mathematical physics – Mechanics – Fluid mechanics – Numerical analysis – Optimization – Probability – Statistics – Mathematical economics – Financial mathematics – Game theory – Mathematical biology – Cryptography – Information theory

Important theorems

:These theorems have interested mathematicians and non-mathematicians alike.

:See list of theorems for more

:Pythagorean theorem – Fermat's last theorem – Gödel's incompleteness theorems – Fundamental theorem of arithmetic – Fundamental theorem of algebra – Fundamental theorem of calculus – Cantor's diagonal argument – Four color theorem – Zorn's lemma – Euler's identity – classification theorems of surfaces – Gauss-Bonnet theorem – Quadratic reciprocity – Riemann-Roch theorem.

Important conjectures
See list of conjectures for more

:These are some of the major unsolved problems in mathematics.

:Goldbach's conjecture – Twin Prime Conjecture – Riemann hypothesis – Poincaré conjecture – Collatz conjecture – P=NP? – open Hilbert problems.

History and the world of mathematicians
See also list of mathematics history topics
:History of mathematics – Timeline of mathematics – Mathematicians – Fields medal – Abel Prize – Millennium Prize Problems (Clay Math Prize) – International Mathematical Union – Mathematics competitions – Lateral thinking – Mathematical abilities and gender issues

Mathematics and other fields
:Mathematics and architecture – Mathematics and education – Mathematics of musical scales

Common misconceptions
Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems.

Pseudomathematics is a form of mathematics-like activity undertaken outside academia, and occasionally by mathematicians themselves. It often consists of determined attacks on famous questions, consisting of proof-attempts made in an isolated way (that is, long papers not supported by previously published theory). The relationship to generally-accepted mathematics is similar to that between pseudoscience and real science. The misconceptions involved are normally based on:

- misunderstanding of the implications of mathematical rigour;

- attempts to circumvent the usual criteria for publication of mathematical papers in a learned journal after peer review, with assumptions of bias;

- lack of familiarity with, and therefore underestimation of, the existing literature.

The case of Kurt Heegner's work shows that the mathematical establishment is neither infallible, nor unwilling to admit error in assessing 'amateur' work. And like astronomy, mathematics owes much to amateur contributors such as Fermat and Mersenne.

Mathematics is not accountancy. Although arithmetic computation is crucial to accountants, their main concern is to verify that computations are correct through a system of doublechecks. Advances in abstract mathematics are mostly irrelevant to the efficiency of concrete bookkeeping, but the use of computers clearly does matter.

Mathematics is not numerology. Numerology uses modular arithmetic to reduce names and dates down to numbers, but assigns emotions or traits to these numbers intuitively or on the basis of traditions.

Mathematical concepts and theorems need not correspond to anything in the physical world. In the case of geometry, for example, it is not relevant to mathematics to know whether points and lines exist in any physical sense, as geometry starts from axioms and postulates about abstract entities called "points" and "lines" that we feed into the system. While these axioms are derived from our perceptions and experience, they are not dependent on them. And yet, mathematics is extremely useful for solving real-world problems. It is this fact that led Eugene Wigner to write an essay on The Unreasonable Effectiveness of Mathematics in the Natural Sciences.

Mathematics is not about unrestricted theorem proving, any more than literature is about the construction of grammatically correct sentences. However, theorems are elements of formal theories, and in some cases computers can generate proofs of these theorems more or less automatically, by means of automated theorem provers. These techniques have proven useful in formal verification of programs and hardware designs. However, they are unlikely to generate (in the near term, at least) mathematics with any widely recognized aesthetic value.

See also

- Mathematical game

- Mathematical problem

- Mathematical puzzle

- Puzzle

Bibliography

- Benson, Donald C., The Moment Of Proof: Mathematical Epiphanies (1999).

- Courant, R. and H. Robbins, What Is Mathematics? (1941);

- Davis, Philip J. and Hersh, Reuben, The Mathematical Experience. Birkhäuser, Boston, Mass., 1980. A gentle introduction to the world of mathematics.

- Boyer, Carl B., History of Mathematics, Wiley, 2nd edition 1998 available, 1st edition 1968 . A concise history of mathematics from the Concept of Number to contemporary Mathematics.

- Gullberg, Jan, Mathematics--From the Birth of Numbers. W.W. Norton, 1996. An encyclopedic overview of mathematics presented in clear, simple language.

- Hazewinkel, Michiel (ed.), Encyclopaedia of Mathematics. Kluwer Academic Publishers 2000. A translated and expanded version of a Soviet math encyclopedia, in ten (expensive) volumes, the most complete and authoritative work available. Also in paperback and on CD-ROM.

- Kline, M., Mathematical Thought from Ancient to Modern Times (1973).

- Pappas, Theoni, The Joy Of Mathematics (1989).

External links

- [http://www.cut-the-knot.org/ Interactive Mathematics Miscellany and Puzzles] — A collection of articles on various math topics, with interactive Java illustrations at cut-the-knot

- Rusin, Dave: [http://www.math-atlas.org/ The Mathematical Atlas]. A guided tour through the various branches of modern mathematics.

- Stefanov, Alexandre: [http://us.geocities.com/alex_stef/mylist.html Textbooks in Mathematics]. A list of free online textbooks and lecture notes in mathematics.

- Weisstein, Eric et al.: [http://www.mathworld.com/ MathWorld: World of Mathematics]. An online encyclopedia of mathematics.

- Polyanin, Andrei: [http://eqworld.ipmnet.ru/ EqWorld: The World of Mathematical Equations]. An online resource focusing on algebraic, ordinary differential, partial differential (mathematical physics), integral, and other mathematical equations.

- A mathematical thesaurus maintained by the [http://nrich.maths.org/ NRICH] project at the University of Cambridge (UK), [http://thesaurus.maths.org/ Connecting Mathematics]

- [http://planetmath.org/ Planet Math]. An online math encyclopedia under construction, focusing on modern mathematics. Uses the GFDL, allowing article exchange with Wikipedia. Uses TeX markup.

- [http://www.mathforge.net/ Mathforge]. A news-blog with topics ranging from popular mathematics to popular physics to computer science and education.

- [http://www.youngmath.net/concerns Young Mathematicians Network (YMN)]. A math-blog "Serving the Community of Young Mathematicians". Topics include: Math News, Grad and Undergrad Life, Job Search, Career, Work & Family, Teaching, Research, Misc...

- [http://metamath.org/ Metamath]. A site and a language, that formalize math from its foundations.

- [http://world.std.com/~reinhold/dir/mathmovies.html Math in the Movies]. A guide to major motion pictures with scenes of real mathematics

- [http://math.cofc.edu/faculty/kasman/MATHFICT/default.html Mathematics in fiction]. Links to works of fiction that refer to mathematics or mathematicians.

- [http://www.mathhelpforum.com/math-help Math Help Forum]. A forum, for math help, math discussion and debate.

- [http://www.sosmath.com/CBB S.O.S. Mathematics Cyberboard] a math help forum which incorporates a LaTeX extension, making it easier for members to write and display math formulae.

- [http://www-history.mcs.st-and.ac.uk/~history/ Mathematician Bibliography]. Extensive history and quotes from all famous mathematicians.

- [http://www.physicsmathforums.com/ Physics Math Forums]

-
Category:School subjects

fiu-vro:Matõmaatiga

zh-min-nan:Sò·-ha̍k

ko:수학

ms:Matematik

ja:数学

simple:Mathematics

th:คณิตศาสตร์

Group (mathematics)
In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. For example, the set of integers is a group under the operation of addition. The branch of mathematics which studies groups is called group theory.

The historical origin of group theory goes back to the works of Évariste Galois (1830), concerning the problem of when an algebraic equation is soluble by radicals. Previous to this work, groups were mainly studied concretely, in the form of permutations; some aspects of abelian group theory were known in the theory of quadratic forms.

A great many of the objects investigated in mathematics turn out to be groups. These include familiar number systems, such as the integers, the rational numbers, the real numbers, and the complex numbers under addition, as well as the non-zero rationals, reals, and complex numbers, under multiplication. Another important example is given by non-singular matrices under multiplication, and more generally, invertible functions under composition. Group theory allows for the properties of these systems and many others to be investigated in a more general setting, and its results are widely applicable. Group theory is also a rich source of theorems in its own right.

Groups underlie many other algebraic structures such as fields and vector spaces. They are also important tools for studying symmetry in all its forms; the principle that the symmetries of any object form a group is foundational for much mathematics. For these reasons, group theory is an important area in modern mathematics, and also one with many applications to mathematical physics (for example, in particle physics).

History

See Group theory.

Basic definitions

A group (G,
- ) is a nonempty set G together with a binary operation
- : G × G → G, satisfying the group axioms below. "a
- b" represents the result of applying the operation
- to the ordered pair (a, b) of elements of G. The group axioms are the following:

- Associativity: For all a, b and c in G, (a
- b)
- c = a
- (b
- c).

- Identity element: There is an element e in G such that for all a in G, e
- a = a
- e = a.

- Inverse element: For all a in G, there is an element b in G such that a
- b = b
- a = e, where e is the identity element from the previous axiom.

You will often also see the axiom

- Closure: For all a and b in G, a
- b belongs to G.
The way that the definition above is phrased, this axiom is not necessary, since binary operations are already required to satisfy closure.
When determining if
- is a group operation, however, it is nonetheless necessary to verify that
- satisfies closure; this is part of verifying that it is in fact a binary operation.

The above axioms are not strictly minimal from a logical viewpoint; they contain a small amount of redundancy. However, the difference is slight and in practice one usually just checks the above axioms.

It should be noted that there is no requirement that the group operation be commutative, that is there may exist elements such that a
- b ≠ b
- a. A group G is said to be abelian (after the mathematician Niels Abel) (or commutative) if for every a, b in G, a
- b = b
- a. Groups lacking this property are called non-abelian.

The order of a group G, denoted by |G| or o(G), is the number of elements of the set G.
A group is called finite if it has finitely many elements, that is if the set G is a finite set.

Note that we often refer to the group (G,
- ) as simply "G", leaving the operation
- unmentioned.
But to be perfectly precise, different operations on the same set define different groups.

Notation for groups

Usually the operation, whatever it really is, is thought of as an analogue of multiplication, and the group operations are therefore written multiplicatively.
That is:

- We write "a · b" or even "ab" for a
- b and call it the product of a and b;

- We write "1" for the identity element and call it the unit element;

- We write "a⁻¹" for the inverse of a and call it the reciprocal of a.

However, sometimes the group operation is thought of as analogous to addition and written additively:

- We write "a + b" for a
- b and call it the sum of a and b;

- We write "0" for the identity element and call it the zero element;

- We write "−a" for the inverse of a and call it the opposite of a.

Usually, only abelian groups are written additively, although abelian groups may also be written multiplicatively. When being noncommittal, one can use the notation (with "
- ") and terminology that was introduced in the definition, using the notation a⁻¹ for the inverse of a.

If S is a subset of G and x an element of G, then, in multiplicative notation, xS is the set of all products ; similarly the notation Sx = ; and for two subsets S and T of G, we write ST for . In additive notation, we write x + S, S + x, and S + T for the respective sets.

Some elementary examples and nonexamples

An abelian group: the integers under addition

A group that we are introduced to in elementary school is the integers under addition.
For this example, let Z be the set of integers, , and let the symbol "+" indicate the operation of addition.
Then (Z,+) is a group (written additively).

Proof:

- If a and b are integers then a + b is an integer. (Closure; + really is a binary operation)

- If a, b, and c are integers, then (a + b) + c = a + (b + c). (Associativity)

- 0 is an integer and for any integer a, 0 + a = a + 0 = a. (Identity element)

- If a is an integer, then there is an integer b := −a, such that a + b = b + a = 0. (Inverse element)

This group is also abelian: a + b = b + a.

The integers with both addition and multiplication together form the more complicated algebraic structure of a ring.
In fact, the elements of any ring form an abelian group under addition, called the additive group of the ring.

Not a group: the integers under multiplication

On the other hand, if we consider the operation of multiplication, denoted by "·", then (Z,·) is not a group:

- If a and b are integers then a · b is an integer. (Closure)

- If a, b, and c are integers, then (a · b) · c = a · (b · c). (Associativity)

- 1 is an integer and for any integer a, 1 · a = a · 1 = a. (Identity element)

- However, it is not true that whenever a is an integer, there is an integer b such that ab = ba = 1. For example, a = 2 is a integer, but the only solution to the equation ab = 1 in this case is b = 1/2. We cannot choose b = 1/2 because 1/2 is not an integer. (Inverse element fails)

Since not every element of (Z,·) has an inverse, (Z,·) is not a group. The most we can say is that it is a commutative monoid.

An abelian group: the nonzero rational numbers under multiplication

Consider the set of rational numbers Q, that is the set of numbers a/b such that
a and b are integers and b is nonzero, and the operation multiplication, denoted by "·".
Since the rational number 0 does not have a multiplicative inverse, (Q,·), like (Z,·), is not a group.

However, if we instead use the set Q \ instead of Q, that is include every rational number except zero, then (Q \ ,·) does form an abelian group (written multiplicatively).
The inverse of a/b is b/a, and the other group axioms are simple to check. We don't lose closure by removing zero, because the product of two nonzero rationals is never zero.

Just as the integers form a ring, the rational numbers form the algebraic structure of a field.
In fact, the nonzero elements of any given field form a group under multiplication, called the multiplicative group of the field.

A finite nonabelian group: permutations of a set

For a more concrete example, consider three colored blocks (red, green, and blue), initially placed in the order RGB. Let a be the action "swap the first block and the second block", and let b be the action "swap the second block and the third block".

field

In multiplicative form, we traditionally write xy for the combined action "first do y, then do x"; so that ab is the action RGB → RBG → BRG, i.e., "take the last block and move it to the front".
If we write e for "leave the blocks as they are" (the identity action), then we can write the six permutations of the set of three blocks as the following actions:

- e : RGB → RGB

- a : RGB → GRB

- b : RGB → RBG

- ab : RGB → BRG

- ba : RGB → GBR

- aba : RGB → BGR

Note that the action aa has the effect RGB → GRB → RGB, leaving the blocks as they were; so we can write aa = e.
Similarly,

- bb = e,

- (aba)(aba) = e, and

- (ab)(ba) = (ba)(ab) = e;
so each of the above actions has an inverse.

By inspection, we can also determine associativity and closure; note for example that

- (ab)a = a(ba) = aba, and

- (ba)b = b(ab) = bab.

This group is called the symmetric group on 3 letters, or S₃.
It has order 6 (or 3 factorial), and is non-abelian (since, for example, ab ≠ ba).
Since S₃ is built up from the basic actions a and b, we say that the set generates it.

Every group can be expressed in terms of permutation groups like S₃; this result is Cayley's theorem and is studied as part of the subject of group actions.

Further examples

For some further examples of groups from a variety of applications, see Examples of groups and List of small groups.

Simple theorems

- A group has exactly one identity element.

- Every element has exactly one inverse.

- You can perform division in groups; that is, given elements a and b of the group G, there is exactly one solution x in G to the equation x
- a = b and exactly one solution y in G to the equation a
- y = b.

- The expression "a₁
- a₂
- ···
- a_n" is unambiguous, because the result will be the same no matter where we place parentheses.

- (Socks and shoes) The inverse of a product is the product of the inverses in the opposite order: (a
- b)⁻¹ = b⁻¹
- a⁻¹.

These and other basic facts that hold for all individual groups form the field of elementary group theory.

Constructing new groups from given ones

#If a subset H of a group (G,
- ) together with the operation
- restricted on H is itself a group, then it is called a subgroup of (G,
- ).
#The direct product of two groups (G,
- ) and (H,•) is the set G×H together with the operation (g₁,h₁)(g₂,h₂) = (g₁
- g₂,h₁•h₂). The product can also be defined with an infinite number of terms.
#The direct external sum of a family of groups is the subgroup of the product constituted by elements that have a finite number of non zero terms. If the family is finite the direct sum and the product are of course the same.
#Given a group G and a normal subgroup N, the quotient group is the set of cosets of G/N together with the operation (gN)(hN)=ghN.

See also

- Glossary of group theory

- Elementary group theory

- List of group theory topics

- Important publications in group theory

- b:Abstract algebra:Groups

Category:Abstract algebra
Category:Group theory
Category:Symmetry

ko:군론

ja:群論

th:กรุป

Finite group
In mathematics, a finite group is a group which has finitely many elements. Some aspects of the theory of finite groups were investigated in great depth in the twentieth century, in particular the local theory, and the theory of solvable groups and nilpotent groups. It is too much to hope for a complete theory: the complexity becomes overwhelming when the group is large.

Less overwhelming, but still of interest, are some of the smaller general linear groups over finite fields. The group theorist [http://www.math.uchicago.edu/~alperin/ J. L. Alperin] has written that "The typical example of a finite group is GL(n,q), the general linear group of n dimensions over the field with q elements. The student who is introduced to the subject with other examples is being completely misled." (Bulletin (New Series) of the American Mathematical Society, 10 (1984) 121) For a discussion of one of the smallest such groups, GL(2,3), see [http://www.log24.com/theory/VisualizingGL2p.html Visualizing GL(2,p)].

Finite groups are directly relevant to symmetry, when that is restricted to a finite number of transformations.
It turns out that continuous symmetry, as modelled by Lie groups, also leads to finite groups, the Weyl groups. In this way, finite groups and their properties can enter centrally in questions, for example in theoretical physics, where their role is not initially obvious.

Every finite group of a prime order is cyclic.

Number of groups for a given set
For each group type (group up to isomorphism) the number of groups for a given underlying set of n elements is n! divided by the order of the automorphism group.

See also

- Lagrange's theorem

- Cauchy's theorem (group theory)

- Sylow theorems

- P-group

- List of small groups

- Character theory

- Representation theory of finite groups

- Modular representation theory

- Classification of finite simple groups

- Monstrous moonshine

- Pro-finite group

- infinite group theory

Category:Group theory

-

ko:유한군

Inverse limit
In mathematics, the inverse limit (also called the projective limit) is a construction which allows one to "glue together" several related objects, the precise manner of the gluing process being specified by morphisms between the objects. Inverse limits can be defined in any category, but we will initially only consider inverse limits of groups.

Formal definition

Algebraic objects

We start with the definition of an inverse system of groups and homomorphisms. Let (I, ≤) be a directed poset (not all authors require I to be directed). Let (A_i)_i∈I be a family of groups and suppose we have a family of homomorphisms f_ij : A_j → A_i for all i ≤ j (note the order) with the following properties:
# f_ii is the identity in A_i,
# f_ik = f_ij O f_jk for all i ≤ j ≤ k.
Then the set of pairs (A_i, f_ij) is called an inverse system of groups and morphisms over I.

We define the inverse limit of the inverse system (A_i, f_ij) as a particular subgroup of the direct product of the A_i's:
: $\varprojlim A_i = \Big\$
The inverse limit, A, comes equipped with natural projections π_i : A → A_i which pick out the ith component of the direct product. The inverse limit and the natural projections satisfy a universal property described in the next section.

This same construction may be carried out if the A_i's are sets, rings, modules (over a fixed ring), algebras (over a fixed field), etc., and the homomorphisms are homomorphisms in the corresponding category. The inverse limit will also belong to that category.

General definition

The inverse limit can be defined abstractly in an arbitrary category by means of a universal property. Let (X_i, f_ij) be an inverse system of objects and morphisms in a category C (same definition as above). The inverse limit of this system is an object X in C together with morphisms π_i : X → X_i (called projections) satisfying π_i = f_ij O π_j . The pair (X, π_i) must be universal in the sense that for any other such pair (Y, ψ_i) there exists a unique morphism u : Y → X making all the "obvious" identities true; i.e. the diagram.

Image:InverseLimit-01.png

must commute for all i, j. The inverse limit is often denoted
: $X = \varprojlim X_i$
with the inverse system (X_i, f_ij) being understood.

Unlike for algebraic objects, the inverse limit may not exist in an arbitrary category. If it does, however, it is unique in a strong sense: given any another inverse limit X′ there exists is a unique isomorphism X′ → X commuting with the projection maps.

We note that an inverse system in category C admits an alternative description in terms of functors. Any partially ordered set I can be considered as a small category where the morphisms consist of arrows i → j iff i ≤ j. An inverse system is then just a contravariant functor I → C.

Examples

- The ring of p-adic integers is the inverse limit of the rings Z/pⁿZ (see modular arithmetic) with the index set being the natural numbers with the usual order, and the morphisms being "take remainder". The natural topology on the p-adic integers is the same as the one described here.

- Pro-finite groups are defined as inverse limits of finite groups.

- Let the index set I of an inverse system (X_i, f_ij) have a greatest element m. Then the natural projection π_m : X → X_m is an isomorphism.

- Inverse limits in the category of topological spaces are given by placing the initial topology on the underlying set-theoretic inverse limit.

- Let (I, =) be the trivial order (not directed). The inverse limit of any corresponding inverse system is just the product.

- Let I consist of three elements i, j, and k with i ≤ j and i ≤ k (not directed). The inverse limit of any corresponding inverse system is the pullback.

Related concepts and generalizations

The categorical dual of an inverse limit is a direct limit (or inductive limit). More general concepts are the limits and colimits of category theory. The terminology is somewhat confusing: inverse limits are limits, while direct limits are colimits.

Category:Category theory
Category:Abstract algebra

Finite group
In mathematics, a finite group is a group which has finitely many elements. Some aspects of the theory of finite groups were investigated in great depth in the twentieth century, in particular the local theory, and the theory of solvable groups and nilpotent groups. It is too much to hope for a complete theory: the complexity becomes overwhelming when the group is large.

Less overwhelming, but still of interest, are some of the smaller general linear groups over finite fields. The group theorist [http://www.math.uchicago.edu/~alperin/ J. L. Alperin] has written that "The typical example of a finite group is GL(n,q), the general linear group of n dimensions over the field with q elements. The student who is introduced to the subject with other examples is being completely misled." (Bulletin (New Series) of the American Mathematical Society, 10 (1984) 121) For a discussion of one of the smallest such groups, GL(2,3), see [http://www.log24.com/theory/VisualizingGL2p.html Visualizing GL(2,p)].

Finite groups are directly relevant to symmetry, when that is restricted to a finite number of transformations.
It turns out that continuous symmetry, as modelled by Lie groups, also leads to finite groups, the Weyl groups. In this way, finite groups and their properties can enter centrally in questions, for example in theoretical physics, where their role is not initially obvious.

Every finite group of a prime order is cyclic.

Number of groups for a given set
For each group type (group up to isomorphism) the number of groups for a given underlying set of n elements is n! divided by the order of the automorphism group.

See also

- Lagrange's theorem

- Cauchy's theorem (group theory)

- Sylow theorems

- P-group

- List of small groups

- Character theory

- Representation theory of finite groups

- Modular representation theory

- Classification of finite simple groups

- Monstrous moonshine

- Pro-finite group

- infinite group theory

Category:Group theory

-

ko:유한군

Discrete space
In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are "isolated" from each other in a certain sense.

Definitions
Given a set X:

- the discrete topology on X is defined by letting every subset of X be open, and X is a discrete topological space if it is equipped with its discrete topology;

- the discrete uniformity on X is defined by letting every superset of the diagonal in X × X be an entourage, and X is a discrete uniform space if it is equipped with its discrete uniformity.

- the discrete metric on X is defined by letting the distance between any distinct points x and y be , and X is a discrete metric space if it is equipped with its discrete metric.
There are many other types of discrete structures that can be placed on a set, but only these cases (to the knowledge of the authors so far of this article) are generally called "spaces".

Properties
The underlying uniformity on a discrete metric space is the discrete uniformity, and the underlying topology on a discrete uniform space is the discrete topology.
Thus, the different notions of discrete space are compatible with one another.
On the other hand, the underlying topology of a non-discrete uniform or metric space can be discrete; an example is the metric space X := (with metric inherited from the real line and given by d(x,y) = |x − y|).
Obviously, this is not the discrete metric; also, this space is not complete and hence not discrete as a uniform space.
Nevertheless, it is discrete as a topological space.
We say that X is topologically discrete but not uniformly discrete or metrically discrete.

Additionally:

- A topological space is discrete if and only if its singletons are open, which is the case if and only if it doesn't contain any accumulation points.

- The singletons form a basis for the discrete topology.

- A uniform space X is discrete if and only if the diagonal is an entourage.

- Every discrete topological space satisfies each of the separation axioms; in particular, every discrete space is Hausdorff, that is, separated.

- A discrete space is compact iff it is finite.

- Every discrete uniform or metric space is complete.

- Combining the above two facts, every discrete uniform or metric space is totally bounded iff it is finite.

- Every discrete metric space is bounded.

- Every discrete space is first-countable, and a discrete space is second-countable iff it is countable.

- Every discrete space is totally disconnected.

- Every non-empty discrete space is second category.

Any function from a discrete topological space to another topological space is continuous, and any function from a discrete uniform space to another uniform space is uniformly continuous.
That is, the discrete space X is free on the set X in the category of topological spaces and continuous maps or in the category of uniform spaces and uniformly continuous maps.
These facts are examples of a much broader phenomenon, in which discrete structures are usually free on sets.

With metric spaces, things are more complicated, because there are several categories of metric spaces, depending on what is chosen for the morphisms.
Certainly the discrete metric space is free when the morphisms are all uniformly continuous maps or all continuous maps, but this says nothing interesting about the metric structure, only the uniform or topological structure.
Categories more relevant to the metric structure can be found by limiting the morphisms to Lipschitz continuous maps or to short maps; however, these categories don't have free objects (on more than one element).
However, the discrete metric space is free in the category of bounded metric spaces and Lipschitz continuous maps, and it is free in the category of metric spaces bounded by 1 and short maps.
That is, any function from a discrete metric space to another bounded metric space is Lipschitz continuous, and any function from a discrete metric space to another metric space bounded by 1 is short.

Going the other direction, a function f from a topological space Y to a discrete space X is continuous if and only it if is locally constant in the sense that every
point in Y has a neighborhood on which f is constant.

Uses

A discrete structure is often used as the "default structure" on a set that doesn't carry any other natural topology, uniformity, or metric.
For example, any group can be considered as a topological group by giving it the discrete topology, implying that theorems about topological groups apply to all groups.
Indeed, analysts may refer to the ordinary, non-topological groups studied by algebraists as "discrete groups" . In some cases, this can be usefully applied, for example combined with Pontryagin duality.

A 0-dimensional manifold (or differentiable or analytical manifold) is nothing but a discrete topological space. In the spirit of the previous paragraph, we can therefore view any discrete group as a 0-dimensional Lie group.

While discrete spaces are not very exciting from a topological viewpoint, one can easily construct interesting spaces from them.
For instance, a product of countably infinitely many copies of the discrete space of natural numbers is homeomorphic to the space of irrational numbers, with the homeomorphism given by the continued fraction expansion. A product of countably infinitely many copies of the discrete space is homeomorphic to the Cantor set; and in fact uniformly homeomorphic to the Cantor set if we use the product uniformity on the product. Such a homeomorphism is given by ternary notation of numbers. (See Cantor space.)

In the foundations of mathematics, the study of compactness properties of products of is central to the topological approach to the ultrafilter principle, which is a weak form of choice.

Indiscrete spaces
In some ways, the opposite of the discrete topology is the trivial topology (also called the indiscrete topology), which has the least possible number of open sets (just the empty set and the space itself). Where the discrete topology is initial or free, the indiscrete topology is final or cofree: every function from a topological space to an indiscrete space is continuous, etc.

Quotation

- Stanislaw Ulam characterized Los Angeles, California as "a discrete space, in which there is an hour's drive between points".

See also

- Cylinder set

Notes

- Stanislaw Ulam's autobiography, Adventures of a Mathematician.

Category:Topology
Category:General topology

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 natural topology induced from that of X called the subspace topology (or the relative topology, or the induced topology).

Definition

Given a topological space $(X, \tau)$ and a subset $S\sube X$ , the subspace topology on $S$ is defined by
: $\tau_S = \lbrace S \cap U \mid U \in \tau \rbrace.$
That is, a subset of $S$ is open in the subspace topology iff it is the intersection of $S$ with an open set in $(X, \tau)$ . If $S$ is equipped with the subspace topology then it is a topological space in its own right, and is called a subspace of $(X, \tau)$ . Subsets of topological spaces are usually assumed to be equipped with the subspace topology unless otherwise stated.

If $S$ is open, closed or dense in $(X, \tau)$ we call $(S, \tau_S)$ an open subspace, closed subspace or dense subspace of $(X, \tau)$ .

Alternatively we can define the subspace topology for a subset $S$ of $X$ as the coarsest topology for which the inclusion map
: $\iota: S \hookrightarrow X$
is continuous.

More generally, suppose $i : S \to X$ is an injection from a set $S$ to a topological space $X$ . Then the subspace topology on $S$ is defined as the coarsest topology for which $i$ is continuous. The open sets in this topology are precisely the ones of the form $i^(U)$ for $U$ open in $X$ . $S$ is then homeomorphic to its image in $X$ (also with the subspace topology) and $i$ is called a topological embedding.

Examples

- Given the real numbers with the usual topology the subspace topology of the natural numbers, as a subspace of the real numbers, is the discrete topology.

- The rational numbers Q considered as a subspace of R do not have the discrete topology (the point 0 is not open in Q).

- Let S = [0,1) be a subspace the real line R. Then [0,½) is open in S but not in R. Likewise [½, 1) is closed in S but not in R. S is both open and closed as a subset of itself but not as a subset of R.

Properties

The subspace topology has the following characteristic property. Let $Y$ be a subspace of $X$ and let $i : Y \to X$ be the inclusion map. Then for any topological space $Z$ a map $f : Z\to Y$ is continuous iff the composite map $i\circ f$ is continuous.
iff
This property is characteristic in the sense that it can be used to define the subspace topology on $Y$ .

We list some further properties of the subspace topology. In the following let $S$ be a subspace of $X$ .

- If $f:X\to Y$ is continuous the restriction to $S$ is continuous.

- If $f:X\to Y$ is continuous then $f:X\to f(X)$ is continuous.

- The closed sets in $S$ are precisely the intersections of $S$ with closed sets in $X$ .

- If $A$ is a subspace of $S$ then $A$ is also a subspace of $X$ with the same topology. In other words the subspace topology that $A$ inherits from $S$ is the same as the one it inherits from $X$ .

- Suppose $S$ is an open subspace of $X$ . Then a subspace of $S$ is open in $S$ iff it is open in $X$ .

- Suppose $S$ is a closed subspace of $X$ . Then a subspace of $S$ is closed in $S$ iff it is closed in $X$ .

- If $B$ is a base for $X$ then $B_S = \$ is a basis for $S$ .

- The topology induced on a subset of a metric space by restricting the metric to this subset coincides with subspace topology for this subset.

Preservation of topological properties

If a topological space has a certain topological property and every subspace shares the same property we say the topological property is hereditary. If only closed subspaces share the property we call it weakly hereditary.

- every open and every closed subspace of a topologically complete space is topologically complete

- every open subspace of a Baire space is a Baire space

- every closed subspace of a compact space is compact.

- being a Hausdorff space is hereditary

- precompactness is hereditary

- being totally disconnected is hereditary

- first countability and second countability is hereditary

References

- Bourbaki, Nicolas, Elements of Mathematics: General Topology, Addison-Wesley (1966)

- Steen, Lynn A. and Seeback, J. Arthur Jr., Counterexamples in Topology, Holt, Rinehart and Winston (1970) ISBN 0030794854.

- Wilard, Stephen. General Topology, Dover Publications (2004) ISBN 0486434796

See also

- the dual notion quotient space

- product topology

- direct sum topology

Category:Topology
Category:General topology

Topological group
In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. Here, G × G is viewed as a topological space by using the product topology. Although we do not do so here, it is common to also require that the topology on G be Hausdorff, namely that any two points in this space have disjoint neighborhoods. The reasons, and some equivalent conditions, are discussed below. In the language of category theory, one would say that topological groups are group objects in the category of topological spaces.

Almost all objects investigated in analysis are topological groups (usually with some additional structure).

Examples

Every group can be trivially made into a topological group by considering it with the discrete topology; such groups are called discrete groups. In this sense, the theory of topological groups subsumes that of ordinary groups.

The real numbers R, together with addition as operation and its ordinary topology, form a topological group. More generally, Euclidean n-space Rⁿ with addition and standard topology is a topological group. More generally still, the additive groups of all topological vector spaces, such as Banach spaces or Hilbert spaces, are topological groups.

The above examples are all abelian. Examples of non-abelian topological groups are given by Lie groups (topological groups that are also manifolds). For instance, the general linear group GL(n,R) of all invertible n-by-n matrices with real entries can be viewed as a topological group with the topology defined by viewing GL(n,R) as a subset of Euclidean space R^n×n. All Lie groups are locally compact.

An example of a topological group which is not a Lie group is given by the rational numbers Q with the topology inherited from R. This is a countable space and it does not have the discrete topology. For a nonabelian example, consider the subgroup of rotations of R³ generated by two rotations by irrational multiples of 2π about different axes.

In every Banach algebra with multiplicative identity, the set of invertible elements forms a topological group under multiplication.

Properties

The algebraic and topological structures of a topological group interact in non-trivial ways. For example, in any topological group the identity component (i.e. the connected component containing the identity element) is a closed normal subgroup.

The inversion operation on a topological group G gives a homeomorphism from G to itself. Likewise, if a is any element of G, then left or right multiplication by a yields a homeomorphism G → G.

Every topological group can be viewed as a uniform space in two ways; the left uniformity turns all left multiplications into uniformly continuous maps while the right uniformity turns all right multiplications into uniformly continuous maps. If G is not abelian, then these two need not coincide. The uniform structures allow to talk about notions such as completeness, uniform continuity and uniform convergence on topological groups.

As a uniform space, every topological group is completely regular. It follows that if a topological group is T₀ (i.e. Kolmogorov), then it is already T₂ (i.e. Hausdorff).

The most natural notion of homomorphism between topological groups is that of a continuous group homomorphism. Topological groups, together with continuous group homomorphisms as morphisms, form a category.

Every subgroup of a topological group is itself a topological group when given the subspace topology. If H is a subgroup of G the set of left or right cosets G/H is a topological space when given the quotient topology (the finest topology on G/H which makes the natural projection q : G → G/H continuous). One can show that the quotient map q : G → G/H is always open.

If H is a normal subgroup of G, then the factor group, G/H becomes a topological group, and the isomorphism theorems known from ordinary group theory remain valid in this setting. However, if H is not closed in the topology of G, then G/H won't be T₀ even if G is. It is therefore natural to restrict oneself to the category of T₀ topological groups, and restrict the definition of normal to normal and closed.

If H is a subgroup of G then the closure of H is also a subgroup. Likewise, if H is a normal subgroup, the closure of H is normal.

Relationship to other areas of mathematics

Of particular importance in harmonic analysis are the locally compact topological groups, because they admit a natural notion of measure and integral, given by the Haar measure. In many ways, the locally compact topological groups serve as a generalization of countable groups, while the compact topological groups can be seen as a generalization of finite groups. The theory of group representations is almost identical for finite groups and for compact topological groups.

See also

- Lie group

- algebraic group

- topological ring

Category:Topological groups

Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T₂ space is a topological space in which points can be separated by neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the Hausdorff condition is the most frequently used and discussed .

Hausdorff spaces are named for Felix Hausdorff, one of the founders of topology. In fact, Hausdorff's original definition of a topological space included the Hausdorff condition as an axiom.

Definitions

Suppose that X is a topological space. Let x and y be points in X. We say that x and y can be separated by neighbourhoods if there exists a neighbourhood U of x and a neighbourhood V of y such that U and V are disjoint (U ∩ V = ).
disjoint

X is a Hausdorff space if any two distinct points of X can be separated by neighborhoods. This is why Hausdorff spaces are also called T₂ spaces or separated spaces.

X is a preregular space if any two topologically distinguishable points can be separated by neighbourhoods. Preregular spaces are also called R₁ spaces.

The relationship between these two conditions is as follows. A topological space is Hausdorff if and only if it is both preregular and Kolmogorov (i.e. distinct points are topologically distinguishable). A topological space is preregular if and only if its Kolmogorov quotient is Hausdorff.

Examples and counterexamples

Almost all spaces encountered in analysis are Hausdorff; most importantly, the real numbers are a Hausdorff space. More generally, all metric spaces are Hausdorff. In fact, many spaces of use in analysis, such as topological groups and topological manifolds, have the Hausdorff condition explicitly stated in their definitions.

A simple example of a topology that is T₁ but is not Hausdorff is the cofinite topology.

Pseudometric spaces typically are not Hausdorff, but they are preregular, and their use in analysis is usually only in the construction of Hausdorff gauge spaces. Indeed, when analysts run across a non-Hausdorff space, it is still probably at least preregular, and then they simply replace it with its Kolmogorov quotient, which is Hausdorff.

In contrast, non-preregular spaces are encountered much more frequently in abstract algebra and algebraic geometry, in particular as the Zariski topology on an algebraic variety or the spectrum of a ring. They also arise in the model theory of intuitionistic logic: every complete Heyting algebra is the algebra of open sets of some topological space, but this space need not be preregular, much less Hausdorff.

Properties

One of the nicest properties of Hausdorff spaces is that limits of sequences, nets, and filters are unique whenever they exist. In fact, a topological space is Hausdorff if and only if every net (or filter) has at most one limit. Similarly, a space is preregular if all of the limits of a given net (or filter) are topologically indistinguishable.

A useful alternative characterization of Hausdorff spaces is the following. A topological space X is Hausdorff iff the diagonal Δ = is closed as a subset of the product space X × X.

Subspaces and products of Hausdorff spaces are Hausdorff, but quotient spaces of Hausdorff spaces need not be Hausdorff. In fact, every topological space can be realized as the quotient of some Hausdorff space.

Hausdorff spaces are T₁, meaning that all singletons are closed. Similarly, preregular spaces are R₀.

Another nice property of Hausdorff spaces is that compact sets are always closed. This may fail for spaces which are non-Hausdorff (there are examples of T₁ spaces where it fails).

The definition of a Hausdorff space says that points can be separated by neighborhoods. It turns out that this implies something which is seemingly stronger: in a Hausdorff space every pair of disjoint compact sets can be separated by neighborhoods. This is an example of the general rule that compact sets often behave like points.

Compactness conditions together with preregularity often imply stronger separation axioms. For example, any locally compact preregular space is completely regular. Compact preregular spaces are normal, meaning that they satisfy Urysohn's lemma and the Tietze extension theorem and have partitions of unity subordinate to locally finite open covers. The Hausdorff versions of these statements are: every locally compact Hausdorff space is Tychonoff, and every compact Hausdorff space is normal Hausdorff.

The following results are some technical properties regarding maps (continuous and otherwise) to and from Hausdorff spaces.

Let f : X → Y be a function and let $\mbox(f) = \$ be its kernel regarded as a subspace of X × X.

- If f is continuous and Y is Hausdorff then ker(f) is closed.

- If f is an open surjection and ker(f) is closed then Y is Hausdorff.

- If f is a continuous, open surjection (i.e. an open quotient map) then Y is Hausdorff iff ker(f) is closed.

If f,g : X → Y are continuous maps and Y is Hausdorff then the equalizer $\mbox(f,g) = \$ is closed in X. It follows that if Y is Hausdorff and f and g agree on a dense subset of X then f = g. In other words, continuous functions into Hausdorff spaces are determined by their values on dense subsets.

Let f : X → Y be a closed surjection such that f⁻¹(y) is compact for all y ∈ Y. Then if X is Hausdorff so is Y.

Let f : X → Y be a quotient map with X a compact Hausdorff space. Then the following are equivalent

- Y is Hausdorff

- f is a closed map

- ker(f) is closed

Preregularity versus regularity

All regular spaces are preregular, as are all Hausdorff spaces. There are many results for topological spaces that hold for both regular and Hausdorff spaces.
Most of the time, these results hold for all preregular spaces; they were listed for regular and Hausdorff spaces separately because the idea of preregular spaces came later.
On the other hand, those results that are truly about regularity generally don't also apply to nonregular Hausdorff spaces.

There are many situations where another condition of topological spaces (such as paracompactness or local compactness) will imply regularity if preregularity is satisfied.
Such conditions often come in two versions: a regular version and a Hausdorff version.
Although Hausdorff spaces aren't generally regular, a Hausdorff space th}}