Topological group

From Wikipedia, the free encyclopedia

 

The real numbers form a topological group under addition

In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a topological structure. Thus, one may perform algebraic operations, because of the group structure, and one may talk about continuous functions, because of the topology.

Topological groups, along with continuous group actions, are used to study continuous symmetries, which have many applications, for example, in physics.

Formal definition

A topological group, G, is a topological space which is also a group such that the group operations of product:

${\displaystyle G\times G\to G:(x,y)\mapsto xy}$

and taking inverses:

${\displaystyle G\to G:x\mapsto x^{-1}}$

are continuous. Here G × G is viewed as a topological space with the product topology.

Although not part of this definition, many authors^[1] require that the topology on G be Hausdorff; it is equivalent to assume that the singleton containing the identity element 1 is a closed subset of G. The reasons, and some equivalent conditions, are discussed below. In any case, any topological group can be made Hausdorff by taking an appropriate canonical quotient.

In the language of category theory, topological groups can be defined concisely as group objects in the category of topological spaces, in the same way that ordinary groups are group objects in the category of sets. Note that the axioms are given in terms of the maps (binary product, unary inverse, and nullary identity), hence are categorical definitions.

Homomorphisms

A homomorphism of topological groups means a continuous group homomorphism G ${\displaystyle \to }$ H. An isomorphism of topological groups is a group isomorphism which is also a homeomorphism of the underlying topological spaces. This is stronger than simply requiring a continuous group isomorphism—the inverse must also be continuous. There are examples of topological groups which are isomorphic as ordinary groups but not as topological groups. Indeed, any non-discrete topological group is also a topological group when considered with the discrete topology. The underlying groups are the same, but as topological groups there is not an isomorphism.

Topological groups, together with their homomorphisms, form a category.

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 with the usual topology form a topological group under addition. More generally, Euclidean n-space Rⁿ is a topological group under addition. Some other examples of abelian topological groups are the circle group S¹, or the torus (S¹)ⁿ for any natural number n.

The classical groups are important examples of non-abelian topological groups. 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 subspace of Euclidean space R^n×n. Another classical group is the orthogonal group O(n), the group of all linear maps from Rⁿ to itself that preserve the length of all vectors. The orthogonal group is compact as a topological space. Much of Euclidean geometry can be viewed as studying the structure of the orthogonal group, or the closely related group O(n) ⋉ Rⁿ of isometries of Rⁿ.

The groups mentioned so far are all Lie groups, meaning that they are smooth manifolds in such a way that the group operations are smooth, not just continuous. Lie groups are the best-understood topological groups; many questions about Lie groups can be converted to purely algebraic questions about Lie algebras and then solved.

An example of a topological group which is not a Lie group is the additive group Q of rational numbers, with the topology inherited from R. This is a countable space, and it does not have the discrete topology. An important example for number theory is the group Z_p of p-adic integers, for a prime number p, meaning the inverse limit of the finite groups Z/pⁿ as n goes to infinity. The group Z_p is well behaved in that it is compact (in fact, homeomorphic to the Cantor set), but it differs from (real) Lie groups in that it is totally disconnected. More generally, there is a theory of p-adic Lie groups, including compact groups such as GL(n,Z_p) as well as locally compact groups such as GL(n,Q_p), where Q_p is the locally compact field of p-adic numbers.

Some topological groups can be viewed as infinite dimensional Lie groups; this phrase is best understood informally, to include several different families of examples. For example, a topological vector space, such as a Banach space or Hilbert space, is an abelian topological group under addition. Some other infinite-dimensional groups that have been studied, with varying degrees of success, are loop groups, Kac–Moody groups, diffeomorphism groups, homeomorphism groups, and gauge groups.

In every Banach algebra with multiplicative identity, the set of invertible elements forms a topological group under multiplication. For example, the group of invertible bounded operators on a Hilbert space arises this way.

Properties

The inversion operation on a topological group G is 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.^[2] If G is not abelian, then these two need not coincide. The uniform structures allow one 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 the identity element is closed in a topological group G, then G is T₂ (Hausdorff), even T_3½ (Tychonoff). If G is not Hausdorff, then one can obtain a Hausdorff group by passing to the quotient group G/K, where K is the closure of the identity.^[3] This is equivalent to taking the Kolmogorov quotient of G.

The Birkhoff–Kakutani theorem states that the following three conditions on a topological group G are equivalent:^[4]

• The identity element 1 is closed in G, and there is a countable basis of neighborhoods for 1 in G.
• G is metrizable (as a topological space).
• There is a left-invariant metric on G that induces the given topology on G. (A metric on G is called left-invariant if for each point a in G, the map x ↦ ax is an isometry from G to itself.)

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 cosets G/H with the quotient topology is called a homogeneous space for G. The quotient map q : G → G/H is always open. For example, for a positive integer n, the sphere Sⁿ is a homogeneous space for the rotation group SO(n+1) in Rⁿ⁺¹, with Sⁿ = SO(n+1)/SO(n). A homogeneous space G/H is Hausdorff if and only if H is closed in G.^[5] Partly for this reason, it is natural to concentrate on closed subgroups when studying topological groups.

Every open subgroup H is also closed in G, since the complement of H is the open set given by the union of open sets gH for g in G \ H.

If H is a normal subgroup of G, then the quotient group G/H becomes a topological group when given the quotient topology. It is Hausdorff if and only if H is closed in G. For example, the quotient group R/Z is isomorphic to the circle group S¹.

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

In any topological group, the identity component (i.e., the connected component containing the identity element) is a closed normal subgroup. If C is the identity component and a is any point of G, then the left coset aC is the component of G containing a. So the collection of all left cosets (or right cosets) of C in G is equal to the collection of all components of G. It follows that the quotient group G/C is totally disconnected.^[6]

The isomorphism theorems from ordinary group theory are not always true in the topological setting. This is because a bijective homomorphism need not be an isomorphism of topological groups. The theorems are valid if one places certain restrictions on the maps involved. For example, the first isomorphism theorem states that if f : G → H is a homomorphism, then the homomorphism from G/ker(f) to im(f) is an isomorphism if and only if the map f is open onto its image.^[7]

Hilbert's fifth problem

There are several strong results on the relation between topological groups and Lie groups. First, every continuous homomorphism of Lie groups G → H is smooth. It follows that a topological group has a unique structure of a Lie group if one exists. Also, Cartan's theorem says that every closed subgroup of a Lie group is a Lie subgroup, in particular a smooth submanifold.

Hilbert's fifth problem asked whether a topological group G which is a topological manifold must be a Lie group. In other words, does G have the structure of a smooth manifold, making the group operations smooth? The answer is yes, by Gleason, Montgomery, and Zippin.^[8] In fact, G has a real analytic structure. Using the smooth structure, one can define the Lie algebra of G, an object of linear algebra which determines a connected group G up to covering spaces. As a result, the solution to Hilbert's fifth problem reduces the classification of topological groups that are topological manifolds to an algebraic problem, albeit a complicated problem in general.

The theorem also has consequences for broader classes of topological groups. First, every compact group (understood to be Hausdorff) is an inverse limit of compact Lie groups. (One important case is an inverse limit of finite groups, called a profinite group. For example, the group Z_p of p-adic integers and the absolute Galois group of a field are profinite groups.) Furthermore, every connected locally compact group is an inverse limit of connected Lie groups.^[9] At the other extreme, a totally disconnected locally compact group always contains a compact open subgroup, which is necessarily a profinite group.^[10] (For example, the locally compact group GL(n,Q_p) contains the compact open subgroup GL(n,Z_p), which is the inverse limit of the finite groups GL(n,Z/p^r) as r goes to infinity.)

Representations of compact or locally compact groups

An action of a topological group G on a topological space X is a group action of G on X such that the corresponding function G × X → X is continuous. Likewise, a representation of a topological group G on a real or complex topological vector space V is a continuous action of G on V such that for each g in G, the map v ↦ gv from V to itself is linear.

Group actions and representation theory are particularly well understood for compact groups, generalizing what happens for finite groups. For example, every finite-dimensional (real or complex) representation of a compact group is a direct sum of irreducible representations. An infinite-dimensional unitary representation of a compact group can be decomposed as a Hilbert-space direct sum of irreducible representations, which are all finite-dimensional; this is part of the Peter–Weyl theorem.^[11] For example, the theory of Fourier series describes the decomposition of the unitary representation of the circle group S¹ on the complex Hilbert space L²(S¹). The irreducible representations of S¹ are all 1-dimensional, of the form z ↦ zⁿ for integers n (where S¹ is viewed as a subgroup of the multiplicative group C*). Each of these representations occurs with multiplicity 1 in L²(S¹).

The irreducible representations of all compact connected Lie groups have been classified. In particular, the character of each irreducible representation is given by the Weyl character formula.

More generally, locally compact groups have a rich theory of harmonic analysis, because they admit a natural notion of measure and integral, given by the Haar measure. Every unitary representation of a locally compact group can be described as a direct integral of irreducible unitary representations. (The decomposition is essentially unique if G is of Type I, which includes the most important examples such as abelian groups and semisimple Lie groups.^[12]) A basic example is the Fourier transform, which decomposes the action of the additive group R on the Hilbert space L²(R) as a direct integral of the irreducible unitary representations of R. The irreducible unitary representations of R are all 1-dimensional, of the form x ↦ e^2πiax for a ∈ R.

The irreducible unitary representations of a locally compact group may be infinite-dimensional. A major goal of representation theory, related to the Langlands classification of admissible representations, is to find the unitary dual (the space of all irreducible unitary representations) for the semisimple Lie groups. The unitary dual is known in many cases such as SL(2,R), but not all.

For a locally compact abelian group G, every irreducible unitary representation has dimension 1. In this case, the unitary dual ${\displaystyle {\hat {G}}}$ is a group, in fact another locally compact abelian group. Pontryagin duality states that for a locally compact abelian group G, the dual of ${\displaystyle {\hat {G}}}$ is the original group G. For example, the dual group of the integers Z is the circle group S¹, while the group R of real numbers is isomorphic to its own dual.

Every locally compact group G has a good supply of irreducible unitary representations; for example, enough representations to distinguish the points of G (the Gelfand–Raikov theorem). By contrast, representation theory for topological groups that are not locally compact has so far been developed only in special situations, and it may not be reasonable to expect a general theory. For example, there are many abelian Banach–Lie groups for which every representation on Hilbert space is trivial.^[13]

Homotopy theory of topological groups

Topological groups are special among all topological spaces, even in terms of their homotopy type. One basic point is that a topological group G determines a path-connected topological space, the classifying space BG (which classifies principal G-bundles over topological spaces, under mild hypotheses). The group G is isomorphic in the homotopy category to the loop space of BG; that implies various restrictions on the homotopy type of G.^[14] Some of these restrictions hold in the broader context of H-spaces.

For example, the fundamental group of a topological group G is abelian. (More generally, the Whitehead product on the homotopy groups of G is zero.) Also, for any field k, the cohomology ring H*(G,k) has the structure of a Hopf algebra. In view of structure theorems on Hopf algebras by Heinz Hopf and Armand Borel, this puts strong restrictions on the possible cohomology rings of topological groups. In particular, if G is a path-connected topological group whose rational cohomology ring H*(G,Q) is finite-dimensional in each degree, then this ring must be a free graded-commutative algebra over Q, that is, the tensor product of a polynomial ring on generators of even degree with an exterior algebra on generators of odd degree.^[15]

In particular, for a connected Lie group G, the rational cohomology ring of G is an exterior algebra on generators of odd degree. Moreover, a connected Lie group G has a maximal compact subgroup K, which is unique up to conjugation, and the inclusion of K into G is a homotopy equivalence. So describing the homotopy types of Lie groups reduces to the case of compact Lie groups. For example, the maximal compact subgroup of SL(2,R) is the circle group SO(2), and the homogeneous space SL(2,R)/SO(2) can be identified with the hyperbolic plane. Since the hyperbolic plane is contractible, the inclusion of the circle group into SL(2,R) is a homotopy equivalence.

Finally, compact connected Lie groups have been classified by Wilhelm Killing, Élie Cartan, and Hermann Weyl. As a result, there is an essentially complete description of the possible homotopy types of Lie groups. For example, a compact connected Lie group of dimension at most 3 is either a torus, the group SU(2) (diffeomorphic to the 3-sphere S³), or its quotient group SU(2)/{±1} ≅ SO(3) (diffeomorphic to RP³).

Generalizations

Various generalizations of topological groups can be obtained by weakening the continuity conditions:^[16]

• A semitopological group is a group G with a topology such that for each c in G the two functions G → G defined by ${\displaystyle x\mapsto xc}$ and ${\displaystyle x\mapsto cx}$ are continuous.
• A quasitopological group is a semitopological group in which the function mapping elements to their inverses is also continuous.
• A paratopological group is a group with a topology such that the group operation is continuous.

Space (mathematics)

From Wikipedia, the free encyclopedia

A hierarchy of mathematical spaces: The inner product induces a norm. The norm induces a metric. The metric induces a topology.

In mathematics, a space is a set (sometimes called a universe) with some added structure.

Mathematical spaces often form a hierarchy, i.e., one space may inherit all the characteristics of a parent space. For instance, all inner product spaces are also normed vector spaces, because the inner product induces a norm on the inner product space such that:

${\displaystyle \left\|x\right\|={\sqrt {\langle x,x\rangle }},}$

where the norm is indicated by enclosing in double vertical lines, and the inner product is indicated enclosing in by angle brackets.

Modern mathematics treats "space" quite differently compared to classical mathematics.

History

Before the golden age of geometry

In the ancient mathematics, "space" was a geometric abstraction of the three-dimensional space observed in everyday life. About 300 BC, Euclid gave axioms for the properties of space. Euclid built all of mathematics on these geometric foundations, going so far as to define numbers by comparing the lengths of line segments to the length of a chosen reference segment.

The method of coordinates (analytic geometry) was adopted by René Descartes in 1637.^[1] At that time, geometric theorems were treated as an absolute objective truth knowable through intuition and reason, similar to objects of natural science;^[2]:11 and axioms were treated as obvious implications of definitions.^[2]:15

Two equivalence relations between geometric figures were used: congruence and similarity. Translations, rotations and reflections transform a figure into congruent figures; homotheties — into similar figures. For example, all circles are mutually similar, but ellipses are not similar to circles. A third equivalence relation, introduced by Gaspard Monge in 1795, occurs in projective geometry: not only ellipses, but also parabolas and hyperbolas, turn into circles under appropriate projective transformations; they all are projectively equivalent figures.

The relation between the two geometries, Euclidean and projective,^[2]:133 shows that mathematical objects are not given to us with their structure.^[2]:21 Rather, each mathematical theory describes its objects by some of their properties, precisely those that are put as axioms at the foundations of the theory.^[2]:20

Distances and angles are never mentioned in the axioms of the projective geometry and therefore cannot appear in its theorems. The question "what is the sum of the three angles of a triangle" is meaningful in the Euclidean geometry but meaningless in projective geometry.

A different situation appeared in the 19th century: in some geometries the sum of the three angles of a triangle is well-defined but different from the classical value (180 degrees). The non-Euclidean hyperbolic geometry, introduced by Nikolai Lobachevsky in 1829 and János Bolyai in 1832 (and Carl Gauss in 1816, unpublished)^[2]:133 stated that the sum depends on the triangle and is always less than 180 degrees. Eugenio Beltrami in 1868 and Felix Klein in 1871 obtained Euclidean "models" of the non-Euclidean hyperbolic geometry, and thereby completely justified this theory.^[2]:24[3]

This discovery forced the abandonment of the pretensions to the absolute truth of Euclidean geometry. It showed that axioms are not "obvious", nor "implications of definitions". Rather, they are hypotheses. To what extent do they correspond to an experimental reality? This important physical problem no longer has anything to do with mathematics. Even if a "geometry" does not correspond to an experimental reality, its theorems remain no less "mathematical truths".^[2]:15

A Euclidean model of a non-Euclidean geometry is a clever choice of some objects existing in Euclidean space and some relations between these objects that satisfy all axioms (therefore, all theorems) of the non-Euclidean geometry. These Euclidean objects and relations "play" the non-Euclidean geometry like contemporary actors playing an ancient performance. Relations between the actors only mimic relations between the characters in the play. Likewise, the chosen relations between the chosen objects of the Euclidean model only mimic the non-Euclidean relations. It shows that relations between objects are essential in mathematics, while the nature of the objects is not.

The golden age and afterwards: dramatic change

According to Nicolas Bourbaki,^[2]:131 the period between 1795 (Geometrie descriptive of Monge) and 1872 (the "Erlangen programme" of Klein) can be called the golden age of geometry. The original space investigated by Euclid is now called three-dimensional Euclidean space. Its axiomatization, started by Euclid 23 centuries ago, was reformed with Hilbert's axioms, Tarski's axioms and Birkhoff's axioms. These axiom systems describe space via primitive notions (such as "point", "between", "congruent") constrained by a number of axioms.

Analytic geometry made great progress, and it succeeded in replacing theorems of classical geometry with computations via invariants of transformation groups.^[2]:134,5 This technique applies simultaneously to many geometries. A definition of space "from scratch", as in Euclid, is now not often used since it does not reveal the relation of this space to other spaces. Since this time, new theorems of classical geometry are of more interest to amateurs rather than to professional mathematicians.^[2]:136 However, the heritage of the classical geometry was not lost. According to Bourbaki,^[2]:138 "passed over in its role as an autonomous and living science, classical geometry is thus transfigured into a universal language of contemporary mathematics".

Simultaneously, numbers began to displace geometry as the foundation of mathematics. For instance, in Richard Dedekind's 1872 essay Stetigkeit und irrationale Zahlen (Continuity and irrational numbers), he asserts that points on a line ought to have the properties of Dedekind cuts, and that therefore a line was the same thing as the set of real numbers. Dedekind is careful to note that this is an assumption that is incapable of being proven. In modern treatments, Dedekind's assertion is often taken to be the definition of a line, thereby reducing geometry to arithmetic. Three-dimensional Euclidean space is defined to be an affine space whose associated vector space of differences of its elements is equipped with an inner product.^[4] Also, a three-dimensional projective space is now defined non-classically, as the space of all one-dimensional subspaces (that is, straight lines through the origin) of a four-dimensional vector space. This shift in foundations requires a new set of axioms, and if these axioms are adopted, the classical axioms for geometry become theorems.

According to the famous inaugural lecture given by Bernhard Riemann in 1854, every mathematical object parametrized by ${\displaystyle n}$ real numbers may be treated as a point of the ${\displaystyle n}$-dimensional space of all such objects.^[2]:140 Contemporary mathematicians follow this idea routinely and find it extremely suggestive to use the terminology of classical geometry nearly everywhere.^[2]:138

An object parametrized by n complex numbers may be treated as a point of a complex n-dimensional space. However, the same object is also parametrized by 2n real numbers (if c is a complex number, then c = a + bi, where a and b are real), thus, a point of a real 2n-dimensional space. The complex dimension differs from the real dimension. This is only the tip of the iceberg. The "algebraic" concept of dimension applies to vector spaces. For topological spaces there are several dimension concepts including inductive dimension and Hausdorff dimension, which can be non-integer (especially for fractals). Some kinds of spaces (for instance, measure spaces) admit no concept of dimension at all.

Functions are important mathematical objects. Usually they form infinite-dimensional function spaces, as noted already by Riemann^[2]:141 and elaborated in the 20th century by functional analysis.

In order to fully appreciate the generality of this approach one should note that mathematics is "a pure theory of forms, which has as its purpose, not the combination of quantities, or of their images, the numbers, but objects of thought" (Hermann Hankel, 1867).^[2]:21 This is a controversial characterization of the purpose of mathematics, which is not necessarily committed to the existence of "objects of thought".

A space consists now of selected mathematical objects (for instance, functions on another space, or subspaces of another space, or just elements of a set) treated as points, and selected relationships between these points. It shows that spaces are just mathematical structures of convenience. One may expect that the structures called "spaces" are more geometric than others, but this is not always true. For example, a differentiable manifold (called also smooth manifold) is much more geometric than a measurable space, but no one calls it "differentiable space" (nor "smooth space").

Taxonomy of spaces

Three taxonomic ranks

Spaces are classified on three levels. Given that each mathematical theory describes its objects by some of their properties, the first question to ask is: which properties?

For example, the upper-level classification distinguishes between Euclidean and projective spaces, since the distance between two points is defined in Euclidean spaces but undefined in projective spaces. These are spaces of different types.

Another example. The question "what is the sum of the three angles of a triangle" makes sense in a Euclidean space but not in a projective space; these are spaces of different types. In a non-Euclidean space the question makes sense but is answered differently, which is not an upper-level distinction.

Also, the distinction between a Euclidean plane and a Euclidean 3-dimensional space is not an upper-level distinction; the question "what is the dimension" makes sense in both cases.

In terms of Bourbaki^[5] the upper-level classification is related to "typical characterization" (or "typification"). However, it is not the same (since two equivalent structures may differ in typification).

On the second level of classification one takes into account answers to especially important questions (among the questions that make sense according to the first level). For example, this level distinguishes between Euclidean and non-Euclidean spaces; between finite-dimensional and infinite-dimensional spaces; between compact and non-compact spaces, etc.

In terms of Bourbaki^[5] the second-level classification is the classification by "species". Unlike biological taxonomy, a space may belong to several species.

On the third level of classification, roughly speaking, one takes into account answers to all possible questions (that make sense according to the first level). For example, this level distinguishes between spaces of different dimension, but does not distinguish between a plane of a three-dimensional Euclidean space, treated as a two-dimensional Euclidean space, and the set of all pairs of real numbers, also treated as a two-dimensional Euclidean space. Likewise it does not distinguish between different Euclidean models of the same non-Euclidean space.

More formally, the third level classifies spaces up to isomorphism. An isomorphism between two spaces is defined as a one-to-one correspondence between the points of the first space and the points of the second space, that preserves all relations between the points, stipulated by the given "typification". Mutually isomorphic spaces are thought of as copies of a single space. If one of them belongs to a given species then they all do.

The notion of isomorphism sheds light on the upper-level classification. Given a one-to-one correspondence between two spaces of the same type, one may ask whether it is an isomorphism or not. This question makes no sense for two spaces of different type.

Isomorphisms to itself are called automorphisms. Automorphisms of a Euclidean space are motions and reflections. Euclidean space is homogeneous in the sense that every point can be transformed into every other point by some automorphism.

Two relations between spaces, and a property of spaces

Topological notions (continuity, convergence, open sets, closed sets etc.) are defined naturally in every Euclidean space. In other words, every Euclidean space is also a topological space. Every isomorphism between two Euclidean spaces is also an isomorphism between the corresponding topological spaces (called "homeomorphism"), but the converse is wrong: a homeomorphism may distort distances. In terms of Bourbaki,^[5] "topological space" is an underlying structure of the "Euclidean space" structure. Similar ideas occur in category theory: the category of Euclidean spaces is a concrete category over the category of topological spaces; the forgetful (or "stripping") functor maps the former category to the latter category.

A three-dimensional Euclidean space is a special case of a Euclidean space. In terms of Bourbaki,^[5] the species of three-dimensional Euclidean space is richer than the species of Euclidean space. Likewise, the species of compact topological space is richer than the species of topological space.

Euclidean axioms leave no freedom, they determine uniquely all geometric properties of the space. More exactly: all three-dimensional Euclidean spaces are isomorphic. In this sense we have "the" three-dimensional Euclidean space. In terms of Bourbaki, the corresponding theory is univalent. In contrast, topological spaces are generally non-isomorphic, their theory is multivalent. A similar idea occurs in mathematical logic: a theory is called categorical if all its models of the same cardinality are isomorphic. According to Bourbaki,^[6] the study of multivalent theories is the most striking feature which distinguishes modern mathematics from classical mathematics.

Types of spaces

Overview of types of abstract spaces. An arrow from space A to space B implies that space A is also a kind of space B. That means, for instance, that a normed vector space is also a metric space.

Linear and topological spaces

Two basic spaces are linear spaces (also called vector spaces) and topological spaces.

Linear spaces are of algebraic nature; there are real linear spaces (over the field of real numbers), complex linear spaces (over the field of complex numbers), and more generally, linear spaces over any field. Every complex linear space is also a real linear space (the latter underlies the former), since each real number is also a complex number.^{[details 1]} Linear operations, given in a linear space by definition, lead to such notions as straight lines (and planes, and other linear subspaces); parallel lines; ellipses (and ellipsoids). However, orthogonal (perpendicular) lines cannot be defined, and circles cannot be singled out among ellipses. The dimension of a linear space is defined as the maximal number of linearly independent vectors or, equivalently, as the minimal number of vectors that span the space; it may be finite or infinite. Two linear spaces over the same field are isomorphic if and only if they are of the same dimension.

Topological spaces are of analytic nature. Open sets, given in a topological space by definition, lead to such notions as continuous functions, paths, maps; convergent sequences, limits; interior, boundary, exterior. However, uniform continuity, bounded sets, Cauchy sequences, differentiable functions (paths, maps) remain undefined. Isomorphisms between topological spaces are traditionally called homeomorphisms; these are one-to-one correspondences continuous in both directions. The open interval ${\displaystyle (0,1)}$ is homeomorphic to the whole real line ${\displaystyle (-\infty ,\infty )}$ but not homeomorphic to the closed interval ${\displaystyle [0,1]}$, nor to a circle. The surface of a cube is homeomorphic to a sphere (the surface of a ball) but not homeomorphic to a torus. Euclidean spaces of different dimensions are not homeomorphic, which seems evident, but is not easy to prove. Dimension of a topological space is difficult to define; "inductive dimension" and "Lebesgue covering dimension" are used. Every subset of a topological space is itself a topological space (in contrast, only linear subsets of a linear space are linear spaces). Arbitrary topological spaces, investigated by general topology (called also point-set topology) are too diverse for a complete classification (up to homeomorphism). They are inhomogeneous (in general). Compact topological spaces are an important class of topological spaces ("species" of this "type"). Every continuous function is bounded on such space. The closed interval ${\displaystyle [0,1]}$ and the extended real line ${\displaystyle [-\infty ,\infty ]}$ are compact; the open interval ${\displaystyle (0,1)}$ and the line ${\displaystyle (-\infty ,\infty )}$ are not. Geometric topology investigates manifolds (another "species" of this "type"); these are topological spaces locally homeomorphic to Euclidean spaces. Low-dimensional manifolds are completely classified (up to homeomorphism).

The two structures discussed above (linear and topological) are both underlying structures of the "linear topological space" structure. That is, a linear topological space is both a linear (real or complex) space and a (homogeneous, in fact) topological space. However, an arbitrary combination of these two structures is generally not a linear topological space; the two structures must conform, namely, the linear operations must be continuous.

Every finite-dimensional (real or complex) linear space is a linear topological space in the sense that it carries one and only one topology that makes it a linear topological space. The two structures, "finite-dimensional (real or complex) linear space" and "finite-dimensional linear topological space", are thus equivalent, that is, mutually underlying. Accordingly, every invertible linear transformation of a finite-dimensional linear topological space is a homeomorphism. In the infinite dimension, however, different topologies conform to a given linear structure, and invertible linear transformations are generally not homeomorphisms.

Affine and projective spaces

It is convenient to introduce affine and projective spaces by means of linear spaces, as follows. An ${\displaystyle n}$-dimensional linear subspace of an ${\displaystyle (n+1)}$-dimensional linear space, being itself an ${\displaystyle n}$-dimensional linear space, is not homogeneous; it contains a special point, the origin. Shifting it by a vector external to it, one obtains an ${\displaystyle n}$-dimensional affine space. It is homogeneous. In the words of John Baez, "an affine space is a vector space that's forgotten its origin". A straight line in the affine space is, by definition, its intersection with a two-dimensional linear subspace (plane through the origin) of the ${\displaystyle (n+1)}$-dimensional linear space. Every linear space is also an affine space.

Every point of the affine space is its intersection with a one-dimensional linear subspace (line through the origin) of the ${\displaystyle (n+1)}$-dimensional linear space. However, some one-dimensional subspaces are parallel to the affine space; in some sense, they intersect it at infinity. The set of all one-dimensional linear subspaces of an ${\displaystyle (n+1)}$-dimensional linear space is, by definition, an ${\displaystyle n}$-dimensional projective space. Choosing an ${\displaystyle n}$-dimensional affine space as before one observes that the affine space is embedded as a proper subset into the projective space. However, the projective space itself is homogeneous. A straight line in the projective space, by definition, corresponds to a two-dimensional linear subspace of the ${\displaystyle (n+1)}$-dimensional linear space.

Defined this way, affine and projective spaces are of algebraic nature; they can be real, complex, and more generally, over any field.

Every real (or complex) affine or projective space is also a topological space. An affine space is a non-compact manifold; a projective space is a compact manifold.

Metric and uniform spaces

Distances between points are defined in a metric space. Every metric space is also a topological space. Bounded sets and Cauchy sequences are defined in a metric space (but not just in a topological space). Isomorphisms between metric spaces are called isometries. A metric space is called complete if all Cauchy sequences converge. Every incomplete space is isometrically embedded into its completion. Every compact metric space is complete; the real line is non-compact but complete; the open interval ${\displaystyle (0,1)}$ is incomplete.

A topological space is called metrizable, if it underlies a metric space. All manifolds are metrizable.

Every Euclidean space is also a complete metric space. Moreover, all geometric notions immanent to a Euclidean space can be characterized in terms of its metric. For example, the straight segment connecting two given points ${\displaystyle A}$ and ${\displaystyle C}$ consists of all points ${\displaystyle B}$ such that the distance between ${\displaystyle A}$ and ${\displaystyle C}$ is equal to the sum of two distances, between ${\displaystyle A}$ and ${\displaystyle B}$ and between ${\displaystyle B}$ and ${\displaystyle C}$.

Uniform spaces do not introduce distances, but still allow one to use uniform continuity, Cauchy sequences, completeness and completion. Every uniform space is also a topological space. Every linear topological space (metrizable or not) is also a uniform space. More generally, every commutative topological group is also a uniform space. A non-commutative topological group, however, carries two uniform structures, one left-invariant, the other right-invariant. Linear topological spaces are complete in finite dimension but generally incomplete in infinite dimension.

Normed, Banach, inner product, and Hilbert spaces

Vectors in a Euclidean space are a linear space, but each vector ${\displaystyle x}$ has also a length, in other words, norm, ${\displaystyle \|x\|}$. A (real or complex) linear space endowed with a norm is a normed space. Every normed space is both a linear topological space and a metric space. A Banach space is a complete normed space. Many spaces of sequences or functions are infinite-dimensional Banach spaces.

The set of all vectors of norm less than one is called the unit ball of a normed space. It is a convex, centrally symmetric set, generally not an ellipsoid; for example, it may be a polygon (on the plane). The parallelogram law (called also parallelogram identity) ${\displaystyle \|x-y\|^{2}+\|x+y\|^{2}=2\|x\|^{2}+2\|y\|^{2}}$ generally fails in normed spaces, but holds for vectors in Euclidean spaces, which follows from the fact that the squared Euclidean norm of a vector is its inner product to itself.

An inner product space is a (real or complex) linear space endowed with a bilinear (or sesquilinear) form satisfying some conditions and called inner product. Every inner product space is also a normed space. A normed space underlies an inner product space if and only if it satisfies the parallelogram law, or equivalently, if its unit ball is an ellipsoid. Angles between vectors are defined in inner product spaces. A Hilbert space is defined as a complete inner product space. (Some authors insist that it must be complex, others admit also real Hilbert spaces.) Many spaces of sequences or functions are infinite-dimensional Hilbert spaces. Hilbert spaces are very important for quantum theory.^[7]

All ${\displaystyle n}$-dimensional real inner product spaces are mutually isomorphic. One may say that the ${\displaystyle n}$-dimensional Euclidean space is the ${\displaystyle n}$-dimensional real inner product space that's forgotten its origin.

Smooth and Riemannian manifolds (spaces)

Smooth manifolds are not called "spaces", but could be. Smooth (differentiable) functions, paths, maps, given in a smooth manifold by definition, lead to tangent spaces. Every smooth manifold is a (topological) manifold. Smooth surfaces in a finite-dimensional linear space (like the surface of an ellipsoid, not a polytope) are smooth manifolds. Every smooth manifold can be embedded into a finite-dimensional linear space. A smooth path in a smooth manifold has (at every point) the tangent vector, belonging to the tangent space (attached to this point). Tangent spaces to an ${\displaystyle n}$-dimensional smooth manifold are ${\displaystyle n}$-dimensional linear spaces. A smooth function has (at every point) the differential, – a linear functional on the tangent space. Real (or complex) finite-dimensional linear, affine and projective spaces are also smooth manifolds.

A Riemannian manifold, or Riemann space, is a smooth manifold whose tangent spaces are endowed with inner product (satisfying some conditions). Euclidean spaces are also Riemann spaces. Smooth surfaces in Euclidean spaces are Riemann spaces. A hyperbolic non-Euclidean space is also a Riemann space. A curve in a Riemann space has the length. A Riemann space is both a smooth manifold and a metric space; the length of the shortest curve is the distance. The angle between two curves intersecting at a point is the angle between their tangent lines.

Waiving positivity of inner product on tangent spaces one gets pseudo-Riemann (especially, Lorentzian) spaces very important for general relativity.

Measurable, measure, and probability spaces

Waiving distances and angles while retaining volumes (of geometric bodies) one moves toward measure theory. Besides the volume, a measure generalizes area, length, mass (or charge) distribution, and also probability distribution, according to Andrey Kolmogorov's approach to probability theory.

A "geometric body" of classical mathematics is much more regular than just a set of points. The boundary of the body is of zero volume. Thus, the volume of the body is the volume of its interior, and the interior can be exhausted by an infinite sequence of cubes. In contrast, the boundary of an arbitrary set of points can be of non-zero volume (an example: the set of all rational points inside a given cube). Measure theory succeeded in extending the notion of volume (or another measure) to a vast class of sets, so-called measurable sets. Indeed, non-measurable sets almost never occur in applications, but anyway, the theory must restrict itself to measurable sets (and functions).

Measurable sets, given in a measurable space by definition, lead to measurable functions and maps. In order to turn a topological space into a measurable space one endows it with a σ-algebra. The σ-algebra of Borel sets is most popular, but not the only choice (Baire sets, universally measurable sets etc. are used sometimes). Alternatively, a σ-algebra can be generated by a given collection of sets (or functions) irrespective of any topology. Quite often, different topologies lead to the same σ-algebra (for example, the norm topology and the weak topology on a separable Hilbert space). Every subset of a measurable space is itself a measurable space.

Standard measurable spaces (called also standard Borel spaces) are especially useful. Every Borel set (in particular, every closed set and every open set) in a Euclidean space (and more generally, in a complete separable metric space) is a standard measurable space. All uncountable standard measurable spaces are mutually isomorphic.

A measure space is a measurable space endowed with a measure. A Euclidean space with Lebesgue measure is a measure space. Integration theory defines integrability and integrals of measurable functions on a measure space.

Sets of measure 0, called null sets, are negligible. Accordingly, a ${\displaystyle {\bmod {0}}}$ isomorphism is defined as isomorphism between subsets of full measure (that is, with negligible complement).

A probability space is a measure space such that the measure of the whole space is equal to 1. The product of any family (finite or not) of probability spaces is a probability space. In contrast, for measure spaces in general, only the product of finitely many spaces is defined. Accordingly, there are many infinite-dimensional probability measures (especially, Gaussian measures), but no infinite-dimensional Lebesgue measure.

Standard probability spaces are especially useful. Every probability measure on a standard measurable space leads to a standard probability space. The product of a sequence (finite or not) of standard probability spaces is a standard probability space. All non-atomic standard probability spaces are mutually isomorphic ${\displaystyle {\bmod {0}};}$ one of them is the interval ${\displaystyle (0,1)}$ with Lebesgue measure.

These spaces are less geometric. In particular, the idea of dimension, applicable (in one form or another) to all other spaces, does not apply to measurable, measure and probability spaces.

A topological space becomes also a measurable space when endowed with the Borel σ-algebra.^{[details 2]} However, the topology is not uniquely determined by its Borel σ-algebra; and not every σ-algebra is the Borel σ-algebra of some topology.^{[details 3]}

Non-commutative geometry

The theoretical study of calculus, known as mathematical analysis, led in the early 20th century to the consideration of linear spaces of real-valued or complex-valued functions. The earliest examples of these were function spaces, each one adapted to its own class of problems. These examples shared many common features, and these features were soon abstracted into Hilbert spaces, Banach spaces, and more general topological vector spaces. These were a powerful toolkit for the solution of a wide range of mathematical problems.

The most detailed information was carried by a class of spaces called Banach algebras. These are Banach spaces together with a continuous multiplication operation. An important early example was the Banach algebra of essentially bounded measurable functions on a measure space X. This set of functions is a Banach space under pointwise addition and scalar multiplication. With the operation of pointwise multiplication, it becomes a special type of Banach space, one now called a commutative von Neumann algebra. Pointwise multiplication determines a representation of this algebra on the Hilbert space of square integrable functions on X. An early observation of von Neumann was that this correspondence also worked in reverse: Given some mild technical hypotheses, a commutative von Neumann algebra together with a representation on a Hilbert space determines a measure space, and these two constructions (of a von Neumann algebra plus a representation and of a measure space) were mutually inverse.

von Neumann then proposed that non-commutative von Neumann algebras should have geometric meaning, just as commutative von Neumann algebras do. Together with Francis Murray, he produced a classification of von Neumann algebras. The direct integral construction shows how to break any von Neumann algebra into a collection of simpler algebras called factors. von Neumann and Murray classified factors into three types. Type I was nearly identical to the commutative case. Types II and III exhibited new phenomena. A type II von Neumann algebra determined a geometry with the peculiar feature that the dimension could be any non-negative real number, not just an integer. Type III algebras were those that were neither types I nor II, and after several decades of effort, these were proven to be closely related to type II factors.

A slightly different approach to the geometry of function spaces developed at the same time as von Neumann and Murray's work on the classification of factors. This approach is the theory of C*-algebras. Here, the motivating example is the C*-algebra ${\displaystyle C_{0}(X)}$, where X is a locally compact Hausdorff topological space. By definition, this is the algebra of continuous complex-valued functions on X that vanish at infinity (which loosely means that the farther you go from a chosen point, the closer the function gets to zero) with the operations of pointwise addition and multiplication. The Gelfand–Naimark theorem implied that there is a correspondence between commutative C*-algebras and geometric objects: Every commutative C*-algebra is of the form ${\displaystyle C_{0}(X)}$ for some locally compact Hausdorff X. The non-commutative C*-algebras, therefore, can be interpreted as non-commutative spaces, much like non-commutative von Neumann algebras.

Both of these examples are now cases of a field called non-commutative geometry. The specific examples of von Neumann algebras and C*-algebras are known as non-commutative measure theory and non-commutative topology, respectively. Non-commutative geometry is not merely a pursuit of generality for its own sake and is not just a curiosity. Non-commutative spaces arise naturally, even inevitably, from some constructions. For example, consider the non-periodic Penrose tilings of the plane by kites and darts. It is a theorem that, in such a tiling, every finite patch of kites and darts appears infinitely often. As a consequence, there is no way to distinguish two Penrose tilings by looking at a finite portion. This makes it impossible to assign the set of all tilings a topology in the traditional sense. Despite this, the Penrose tilings determine a non-commutative C*-algebra, and consequently they can be studied by the techniques of non-commutative geometry. Another example, and one of great interest within differential geometry, comes from foliations of manifolds. These are ways of splitting the manifold up into smaller-dimensional submanifolds called leaves, each of which is locally parallel to others nearby. The set of all leaves can be made into a topological space. However, the example of an irrational rotation shows that this topological space can be bizarre and the techniques of classical measure theory may be useless on it. However, there is a non-commutative von Neumann algebra associated to the leaf space of a foliation, and once again, this gives an otherwise unintelligible space a good geometric structure.

Schemes

Algebraic geometry studies the geometric properties of polynomial equations. Polynomials are a type of function defined by the basic arithmetic operations of addition and multiplication. Because of this, they are closely tied to algebra. Algebraic geometry offers a way to apply geometric techniques to questions of pure algebra, and vice versa.

The type of space that underlies most modern algebraic geometry was introduced by Alexander Grothendieck and is called a scheme. One of the building blocks of a scheme is a topological space. Topological spaces have continuous functions, but continuous functions are too general to reflect the underlying algebraic structure of interest. The other ingredient in a scheme, therefore, is a sheaf on the topological space, called the "structure sheaf". On each open subset of the topological space, the sheaf specifies a collection of functions, called "regular functions". The topological space and the structure sheaf together are required to satisfy conditions that mean the functions come from algebraic operations.

Like manifolds, schemes are defined as spaces which are locally modeled on a familiar space. In the case of manifolds, the familiar space is Euclidean space. For a scheme, the local models are called affine schemes. Affine schemes provide a direct link between algebraic geometry and commutative algebra. The fundamental objects of study in commutative algebra are commutative rings. If R is a commutative ring, then there is a corresponding affine scheme ${\displaystyle \operatorname {Spec} R}$ which translates the algebraic structure of R into geometry. Conversely, every affine scheme determines a commutative ring, the global sections of its structure sheaf. These two operations are mutually inverse, so affine schemes provide a new language with which to study questions in commutative algebra. By definition, every point in a scheme has an open neighborhood which is an affine scheme.

There are many schemes which are not affine. Often, this is an unavoidable consequence of the geometry of the scheme. The most important example of this is projective space. Projective space satisfies a condition called properness which is analogous to compactness. Affine schemes cannot be proper (except in trivial situations like when the scheme has only a single point), and hence no projective space is an affine scheme (except for zero-dimensional projective spaces). Projective space is closely related to the theory of perspective and to homogeneous polynomials. Projective schemes, meaning those that arise as closed subschemes of a projective space, are the single most important family of schemes.^[8]

Several generalizations of schemes have been introduced. Michael Artin defined an algebraic space to be an object which is the quotient of a scheme by certain types of equivalence relations, specifically, equivalence relations which define étale morphisms. Algebraic spaces retain many of the useful properties of schemes while simultaneously being more flexible. For instance, the Keel–Mori theorem can be used to show that many moduli spaces are algebraic spaces.

More general than an algebraic space is a Deligne–Mumford stack. DM stacks are similar to schemes, but they permit singularities that cannot be described solely in terms of polynomials. They play the same role for schemes that orbifolds do for manifolds. For example, the quotient of the affine plane by a finite group of rotations around the origin yields a Deligne–Mumford stack that is not a scheme or an algebraic space. Away from the origin, the quotient by the group action identifies finite sets of equally spaced points on a circle. But at the origin, the circle consists of only a single point, the origin itself, and the group action fixes this point. In the quotient DM stack, however, this point comes with the extra data of being a quotient. This kind of refined structure is useful in the theory of moduli spaces, and in fact, it was originally introduced to describe moduli of algebraic curves.

A yet further generalization are the algebraic stacks, also called Artin stacks. DM stacks are limited to quotients by finite group actions. While this suffices for many problems in moduli theory, it is too restrictive for others. Artin stacks permit more general quotients, and hence more moduli problems can be treated using Artin stacks than DM stacks.

Topoi

In Grothendieck's work on the Weil conjectures, he introduced a new type of topology now called a Grothendieck topology. A topological space (in the ordinary sense) axiomatizes the notion of "nearness," making two points be nearby if and only if they lie in many of the same open sets. By contrast, a Grothendieck topology axiomatizes the notion of "covering." A covering of a space is a collection of subspaces that jointly contain all the information of the ambient space. Since sheaves are defined in terms of coverings, a Grothendieck topology can also be seen as an axiomatization of the theory of sheaves.

Grothendieck's work on his topologies led him to the theory of topoi, which he believed to be one of his greatest mathematical ideas. A sheaf (either on a topological space or with respect to a Grothendieck topology) is used to express local data. The category of all sheaves carries all possible ways of expressing local data. Since topological spaces are constructed from points, which are themselves a kind of local data, the category of sheaves can therefore be used as a replacement for the original space. Grothendieck consequently defined a topos to be a category of sheaves and studied topoi as objects of interest in their own right. These are now called Grothendieck topoi.

Every topological space determines a topos, and vice versa. There are topological spaces where taking the associated topos loses information, but these are generally considered pathological. (A necessary and sufficient condition is that the topological space be a sober space.) Conversely, there are topoi whose associated topological spaces do not capture the original topos. But, far from being pathological, these topoi can be of great mathematical interest. For instance, Grothendieck's theory of étale cohomology (which eventually led to the proof of the Weil conjectures) can be phrased as cohomology in the étale topos of a scheme, and this topos does not come from a topological space.

Topological spaces in fact lead to very special topoi called locales. The set of open subsets of a topological space determines a lattice. The axioms for a topological space cause these lattices to be complete Heyting algebras. The theory of locales takes this as its starting point. A locale is defined to be a complete Heyting algebra, and the elementary properties of topological spaces are re-expressed and reproved in these terms. The concept of a locale turns out to be more general than a topological space, in that every sober topological space determines a unique locale, but many interesting locales do not come from topological spaces. Because locales need not have points, the study of locales is somewhat jokingly called pointless topology.

Topoi also display deep connections to mathematical logic. Every Grothendieck topos has a special sheaf called a subobject classifier. This subobject classifier functions like the set of all possible truth values. In the topos of sets, the subobject classifier is the set ${\displaystyle \{0,1\}}$, corresponding to "False" and "True". But in other topoi, the subobject classifier can be much more complicated. Lawvere and Tierney recognized that axiomatizing the subobject classifier yielded a more general kind of topos, now known as an elementary topos, and that elementary topoi were models of intuitionistic logic. In addition to providing a powerful way to apply tools from logic to geometry, this made possible the use of geometric methods in logic.