topological space

Given a metric space (X,d) we call a point p close to a set A if d(p,A) = 0 Similarly a set B is called close to a set A if See also:

Closeness mathematics, Closeness mathematics - Definition, Closeness mathematics - Properties, Closeness mathematics - Closeness relation between a point and a set, Closeness mathematics - Closeness relation between two sets, Closeness mathematics - Generalized definition

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There may be different notions of connectedness that are intuitively similar, but different as formally defined concepts. We might wish to call a topological space connected if each pair of points in it is joined by a path. However this concept turns out to be different from standard topological connectedness; in particular, there are connected topological spaces for which this property does not hold. Because of this, different terminology is used; ...

See also:

Connectedness, Connectedness - Other notions of connectedness, Connectedness - Connectivity

Read more here: » Connectedness: Encyclopedia II - Connectedness - Other notions of connectedness

In mathematics, convergence describes limiting behaviour, particularly of an infinite sequence or series toward some limit. To assert convergence is to claim the existence of such a limit, which may be itself unknown. For any fixed standard of accuracy, however, you can always be sure to be within that limit, provided you have gone far enough. The following lists more specific usages of this word: Convergent series provides a general mathematical definition and a context in which to understand the remaining ...

See also:

Convergence, Convergence - Science fiction and popular culture, Convergence - Mathematics, Convergence - Natural sciences, Convergence - Computing and technology, Convergence - Social sciences, Convergence - Political parties

Read more here: » Convergence: Encyclopedia II - Convergence - Mathematics

In topology, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More formally, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary of S is called a boundary point of S. Notations used for boundary of a set S include bd(S), fr(S), and . There are several common (and equivalent) definitions to the boundary of ...

Including:

• Boundary topology - Examples
• Boundary topology - Properties
• Boundary topology - Boundary of a boundary

Read more here: » Boundary topology: Encyclopedia - Boundary topology

In topology, a CW complex is a type of topological space introduced by J.H.C. Whitehead to meet the needs of homotopy theory. The idea was to have a class of spaces that was broader than simplicial complexes (we could say now, had better categorical properties); but still retained a combinatorial nature, so that computational considerations were not ignored. The name itself is unrevealing: C ...

Including:

• CW complex - Attaching cells
• CW complex - CW complexes are defined inductively
• CW complex - 'The' homotopy category
• CW complex - Properties

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Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. It is half-jokingly known as "generalized abstract nonsense". Categories appear in most branches of mathematics, in some areas of theoretical computer science and mathematical physics, and have been a unifying notion. Categories were first introduced by Samuel Eilenberg and Saunders Ma ...

Including:

• Category theory - Background
• Category theory - Historical notes
• Category theory - Categories objects and morphisms
• Category theory - Some properties of morphisms
• Category theory - Functors
• Category theory - Natural transformations and isomorphisms
• Category theory - Universal constructions limits and colimits
• Category theory - Equivalent categories
• Category theory - Further concepts and results
• Category theory - Higher-dimensional categories

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In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. Many of the infinite-dimensional function spaces studied in functional analysis are examples of Banach spaces. Banach space - Definition. Banach spaces are defined as complete normed vector spaces. This means that a Banach space is a vector space V over the real or complex numbers with a norm ||.|| such that every Cauchy sequence (with respect to the metric dIncluding:

• Banach space - Definition
• Banach space - Examples
• Banach space - Linear operators
• Banach space - Dual space
• Banach space - Relationship to Hilbert spaces
• Banach space - Derivatives
• Banach space - Generalizations
• Banach space - Literature

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In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec(R), is defined to be the set of all proper prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. Spectrum of a ring - Zariski topology. Spec(R) can be turned into a topological space as follows: a subset V of Spec(R) is closed if and only if there exists a subset I of R< ...

Including:

• Spectrum of a ring - Zariski topology
• Spectrum of a ring - Sheaves and schemes
• Spectrum of a ring - Functoriality
• Spectrum of a ring - Motivation from algebraic geometry
• Spectrum of a ring - External link

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A manifold is a mathematical space which is constructed, like a patchwork, by gluing and bending together copies of simple spaces. For example, a circle can be constructed by bending two line segments into arcs which overlap at their ends and gluing them together where they overlap. The motivation for working with manifolds is that you begin with a relatively simple space which is well understood, and build up a manifold, which may be very complicated, from copies of that simple space. By choosing different spaces as base material, di ...

Including:

• Manifold - Introduction
• Manifold - Motivational example: the circle
• Manifold - Charts atlases and transition maps
• Manifold - Construction
• Manifold - Charts
• Manifold - Patchwork
• Manifold - Zeros of a function
• Manifold - Identifying points of a manifold
• Manifold - Cartesian products
• Manifold - Gluing along boundaries
• Manifold - Topological manifolds
• Manifold - Differentiable manifolds
• Manifold - Orientability
• Manifold - Möbius strip
• Manifold - Klein bottle
• Manifold - Real projective plane
• Manifold - Other types and generalizations of manifolds
• Manifold - History

Read more here: » Manifold: Encyclopedia - Manifold

Infinity is a term with very distinct, separate meanings which arise in theology, philosophy, mathematics and everyday life. Popular or colloquial usage of the term often does not accord with its more technical meanings. The word infinity comes from Latin : "Infinito", unending. In theology, for example in the work of theologians such as Duns Scotus, the infinite nature of God invokes a sense of being without constraint, rather than a sense of being unlimited in quantity. In philosophy, infinity can be attrib ...

Including:

• Infinity - History
• Infinity - Ancient view of infinity
• Infinity - Views from the Renaissance to modern times
• Infinity - Modern philosophical views
• Infinity - Infinity symbol
• Infinity - Mathematical infinity
• Infinity - Infinity in real analysis
• Infinity - Infinity in complex analysis
• Infinity - Arithmetic properties of infinity
• Infinity - Infinity in set theory
• Infinity - Mathematics without infinity
• Infinity - Use of infinity in common speech
• Infinity - Physical infinity
• Infinity - Infinity in cosmology
• Infinity - Three types of infinities
• Infinity - Infinity in science fiction
• Infinity - Note

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In common usage, the dimensions (from Latin "measured out") of an object are the parameters or measurements required to define its shape and size, that is, usually, its height, width, and length. In mathematics, the dimensions of a space are the parameters required to describe a particular object in this space. The dimension of a space is the number of these parameters. For example, locating a city on the Earth requires two parameters: longitude and latitude; the corresponding space has therefore two dimensions an ...

Including:

• Dimension - Physical dimensions
• Dimension - Spatial dimensions
• Dimension - Time
• Dimension - Additional dimensions
• Dimension - Units
• Dimension - Mathematical dimensions
• Dimension - Hamel dimension
• Dimension - Manifolds
• Dimension - Lebesgue covering dimension
• Dimension - Inductive dimension
• Dimension - Hausdorff dimension
• Dimension - Hilbert spaces
• Dimension - Krull dimension of commutative rings
• Dimension - Science fiction
• Dimension - Anaglyph
• Dimension - 3-D film
• Dimension - Modern 3-D films
• Dimension - More dimensions
• Dimension - Degrees of freedom
• Dimension - Other

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In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain a bigger one. A presheaf is similar to a sheaf, but it may not be possible to glue. Sheaves enable one to discuss in a refined way what is a local property, as appl ...

Including:

• Sheaf mathematics - Introduction
• Sheaf mathematics - The formal definition
• Sheaf mathematics - Definition of a presheaf
• Sheaf mathematics - The gluing axiom
• Sheaf mathematics - Examples
• Sheaf mathematics - Morphisms of sheaves
• Sheaf mathematics - Stalks of a sheaf at a point and germs of functions
• Sheaf mathematics - The étale space of a sheaf
• Sheaf mathematics - Generalizations
• Sheaf mathematics - History

Read more here: » Sheaf mathematics: Encyclopedia - Sheaf mathematics

In mathematics, adjoint functors are pairs of functors which stand in a particular relationship with one another. Such functors are ubiquitous in mathematics. Adjoint functors are studied in a branch of mathematics known as category theory. Like much of category theory, the general notion of adjoint functors arises at an abstract level beyond the everyday usage of mathematicians. Adjoint functors can be considered from several different points of view. This article starts with a number of introductory sections considering some ...

Including:

• Adjoint functors - Motivation
• Adjoint functors - Ubiquity of adjoint functors
• Adjoint functors - Deep problems formulated with adjoint functors
• Adjoint functors - Adjoint functors as solving optimization problems
• Adjoint functors - The case of partial orders
• Adjoint functors - Formal definitions
• Adjoint functors - Examples
• Adjoint functors - Properties
• Adjoint functors - Uniqueness of adjoints
• Adjoint functors - Relation to universal constructions
• Adjoint functors - Characterization via unit and co-unit
• Adjoint functors - Adjoints preserve certain limits
• Adjoint functors - Additivity
• Adjoint functors - Composition
• Adjoint functors - Adjoint pairs extend equivalences
• Adjoint functors - General existence theorem

Read more here: » Adjoint functors: Encyclopedia - Adjoint functors

Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. It can be seen as the study of solution sets of systems of algebraic equations. When there is more than one variable, geometric considerations enter and are important to understand the phenomenon. One can say that the subject starts where equation solving leaves off, and it becomes at least as important to understand the totality of solutions of a system of equations as to find som ...

Including:

• Algebraic geometry - Zeroes of simultaneous polynomials
• Algebraic geometry - Affine varieties
• Algebraic geometry - Regular functions
• Algebraic geometry - The category of affine varieties
• Algebraic geometry - Projective space
• Algebraic geometry - The modern viewpoint
• Algebraic geometry - Notes and history

Read more here: » Algebraic geometry: Encyclopedia - Algebraic geometry

Analysis is the generic name given to any branch of mathematics which depends upon the concepts of limits and convergence, and studies closely related topics such as continuity, integration, differentiability and transcendental functions. These topics are often studied in the context of real numbers, complex numbers, and their functions. However, they can also be defined and studied in any space of mathematical objects that is equipped with a definition of "nearness" (a topological space) or "distance" (a metric space). Mathematical analysis ...

Including:

• Mathematical analysis - History
• Mathematical analysis - Subdivisions

Read more here: » Mathematical analysis: Encyclopedia - Mathematical analysis

In mathematics, the closure of a set S consists of all points which are intuitively "close to S". A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior. Closure topology - Definitions. Closure topology - Point of closure. For S a subset of an Euclidean space, x is a point of closure of S if every open ball centered at x contains a point o ...

Including:

• Closure topology - Definitions
• Closure topology - Point of closure
• Closure topology - Limit point
• Closure topology - Closure of a set
• Closure topology - Examples
• Closure topology - Closure operator
• Closure topology - Facts about closures

Read more here: » Closure topology: Encyclopedia - Closure topology

In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. Simple examples are the circle or the straight line. A large number of other curves have been studied in geometry. This article is about the general theory. The term curve is also used in ways making it almost synonymous with mathematical function (as in learning curve), or graph of a function (Phillips curve). Curve - Definitions. In ma ...

Including:

• Curve - Definitions
• Curve - Conventions and terminology
• Curve - Lengths of curves
• Curve - Differential geometry
• Curve - Algebraic curve
• Curve - History

Read more here: » Curve: Encyclopedia - Curve

See Cartesian coordinate system or Coordinates (mathematics) for a more elementary introduction to this topic. In mathematics as applied to geometry, physics or engineering, a coordinate system is a system for assigning a tuple of numbers to each point in an n-dimensional space. "Numbers" in many cases means real numbers, but, depending on context, can mean complex numbers or elements of some other field. If the space or manifold is curved, it may not be possible to provide one consistent coordinate system for the entire space. In this case, a set of coordinate systems, called charts, are ...

Including:

• Coordinate system - Examples
• Coordinate system - Transformations
• Coordinate system - Singularities
• Coordinate system - Systems commonly used
• Coordinate system - Astronomical systems

Read more here: » Coordinate system: Encyclopedia - Coordinate system

In the absence of a more specific context, convergence denotes the approach toward a definite value, as time goes on; or to a definite point, a common view or opinion, or toward a fixed or equilibrium state. Convergence - Science fiction and popular culture. Convergence (goth festival) refers to an annual convention in which goths meet each other in 'real life' rather than online, as is customarily done. CONvergence (convention) is a speculative fiction convention in Minnesota. Harmo ...

Including:

• Convergence - Science fiction and popular culture
• Convergence - Mathematics
• Convergence - Natural sciences
• Convergence - Computing and technology
• Convergence - Social sciences
• Convergence - Political parties

Read more here: » Convergence: Encyclopedia - Convergence

In mathematics, the unit interval is the interval [0,1], that is the set of all real numbers x such that zero is less than or equal to x and x is less than or equal to one. The unit interval plays a fundamental role in homotopy theory, a major branch of topology. It is a metric space, compact, contractible, path connected and locally path connected. As a topological space, it is homeomorphic to the extended real number line. The unit interval is a one-dimensional analytical manifold with boundary {0,1}, carrying a ...

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