* In the cocountable topology on R ( or any uncountable set for that matter ), no infinite set is compact.

The cocountable topology or countable complement topology on any set X consists of the empty set and all cocountable subsets of X, that is all sets whose complement in X is countable.

Every set X with the cocountable topology is Lindelöf, since every nonempty open set omits only countably many points of X.

However, in the cocountable topology all convergent sequences are eventually constant, so limits are unique.

The cocountable topology on a countable set is the discrete topology.

The cocountable topology on an uncountable set is hyperconnected, thus connected, locally connected and pseudocompact, but neither weakly countably compact nor countably metacompact, hence not compact.

" Almost all " is sometimes used synonymously with " all but finitely many " ( formally, a cofinite set ) or " all but a countable set " ( formally, a cocountable set ); see almost.

If the complement is not finite, but it is countable, then one says the set is cocountable.

In mathematics, a cocountable subset of a set X is a subset Y whose complement in X is a countable set.

While the rational numbers are a countable subset of the reals, for example, the irrational numbers are a cocountable subset of the reals.

The set of all subsets of X that are either countable or cocountable forms a σ-algebra, i. e., it is closed under the operations of countable unions, countable intersections, and complementation.

Closely linked to these cohomology theories, he originated topos theory as a generalisation of topology ( relevant also in categorical logic ).

The spaceX * of all linear maps into K ( which is called the algebraic dual space to distinguish it from X ′) also induces a weak topology which is finer than that induced by the continuous dual since X ′ ⊆ X *.

Over any vector space without topology, we may also notate the vectors by kets and the linear functionals by bras.

Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, and combinatorics also has many applications in optimization, computer science, ergodic theory and statistical physics.

Differential topology also deals with questions like these, which specifically pertain to the properties of differentiable mappings on R < sup > n </ sup > ( for example the tangent bundle, jet bundles, the Whitney extension theorem, and so forth ).

Since Euclidean space is a metric space it is also a topological space with the natural topology induced by the metric.

Dyson also did work in a variety of topics in mathematics, such as topology, analysis, number theory and random matrices.

Grothendieck also saw how to phrase the definition of covering abstractly ; this is where the definition of a Grothendieck topology comes from.

Weyl brought the early period of the development of the theory of Lie groups to fruition, for not only did he classify irreducible representations of semisimple Lie groups and connect the theory of groups with quantum mechanics, but he also put Lie's theory itself on firmer footing by clearly enunciating the distinction between Lie's infinitesimal groups ( i. e., Lie algebras ) and the Lie groups proper, and began investigations of topology of Lie groups ( Borel ( 2001 ), ).

Filters appear in order and lattice theory, but can also be found in topology whence they originate.

Separable metrizable spaces can also be characterized as those spaces which are homeomorphic to a subspace of the Hilbert cube, i. e. the countably infinite product of the unit interval ( with its natural subspace topology from the reals ) with itself, endowed with the product topology.

For measure spaces that are also topological spaces various compatibility conditions can be placed for the measure and the topology.

Escher also studied the mathematical concepts of topology.

In mathematics, pointless topology ( also called point-free or pointfree topology ) is an approach to topology that avoids mentioning points.

This topology differs from another, perhaps more obvious, topology called the box topology, which can also be given to a product space and which agrees with the product topology when the product is over only finitely many spaces.

Real analysis is also used as a starting point for other areas of analysis, such as complex analysis, functional analysis, and harmonic analysis, as well as for motivating the development of topology, and as a tool in other areas, such as applied mathematics.

Although they also have a toroidal magnetic field topology, stellarators are distinct from tokamaks in that they are not azimuthally symmetric.

The following definitions are also fundamental to algebraic topology, differential topology and geometric topology.

** In topology, morphisms between topological spaces are called continuous maps, and an automorphism of a topological space is a homeomorphism of the space to itself, or self-homeomorphism ( see homeomorphism group ).

A similar type of investigation occurs in many mathematical theories, such as the study of continuous maps ( morphisms ) between topological spaces in topology ( the associated category is called Top ), and the study of smooth functions ( morphisms ) in manifold theory.

A subset K of a topological space X is called compact if it is compact in the induced topology.

There is only one such topology ; it is called the topology of pointwise convergence.

The metric topology on E < sup > n </ sup > is called the Euclidean topology.

* An alternate topology called the grounded bridge amplifier-pdf

In general, when an n-dimensional grid network is connected circularly in more than one dimension, the resulting network topology is a torus, and the network is called " toroidal ".

If E / F is a Galois extension, then Gal ( E / F ) can be given a topology, called the Krull topology, that makes it into a profinite group.

A category together with a choice of Grothendieck topology is called a site.

For compact space | compact 2-dimensional surfaces without boundary ( topology ) | boundary, if every loop can be continuously tightened to a point, then the surface is topologically Homeomorphism | homeomorphic to a 2-sphere ( usually just called a sphere ).

At the beginning of the 20th century, Henri Poincaré was working on the foundations of topology — what would later be called combinatorial topology and then algebraic topology.

In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology.