compact space

In mathematics and physics, continuous spectrum is, roughly speaking, a non-countable set of eigenvalues of an operator. An operator acting on a Hilbert space is said to have a continuous spectrum if its eigenvalues can be changed continuously. If the spectrum of an operator is not continuous, we say that it is has discrete spectrum. Some of the basic questions in spectral theory are to characterise the discrete spectrum and purely continuous spectrum, just as a measure, such as a probability measure, can typically ...

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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 mathematics, a function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded. In other words, there exists a number M>0 such that for all x in X. The concept should not be confused with that of a bounded operator. An important special case is a bounded sequence, where X is taken to be the set N of natural numbers. Thus a sequence f = ( a0, a1< ...

Including:

• Bounded function - Examples

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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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Field of sets - Sigma algebras and measure spaces. If an algebra over a set is closed under countable intersections and countable unions, it is called a sigma algebra and the corresponding field of sets is called a measureable space. The complexes of a measurable space are called measureable sets. A measure space is a triple where is a measurable space and μ is a measure defined on it. If μ is in fact a probability measure w ...

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Field of sets, Field of sets - Fields of sets in the representation theory of Boolean algebras, Field of sets - Stone representation, Field of sets - Separative and compact fields of sets: towards Stone duality, Field of sets - Fields of sets with additional structure, Field of sets - Sigma algebras and measure spaces, Field of sets - Topological fields of sets, Field of sets - Preorder fields, Field of sets - Complex algebras and fields of sets on relational structures

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Every homeomorphism is open, closed, and continuous. In fact, a bijective continuous map is a homeomorphism iff it's open, or equivalently, iff it's closed. If Y has the discrete topology (i.e. all subsets are open and closed) then every function f : X → Y is both open and closed (but not necessarily continuous). Whenever we have a product of topological spaces X=ΠXi, then the natural projections pi : X → Xi ...

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Open and closed maps, Open and closed maps - Examples, Open and closed maps - Facts and theorems

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The shape of the Universe is not concerned with a geometry near to a dense mass. Rather, the geometries under investigation describe a Universe where the average density of mass is evenly distributed. Notwithstanding that the Universe is "weakly" inhomogeneous and anisotropic in the large-scale structure of the cosmos, both astronomical and cosmological measurements determine the observable Universe to be, on average, homogeneous, isotropic, and expanding or accelerating. Although the geometries within the observable Universe are generated b ...

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Shape of the Universe, Shape of the Universe - Description, Shape of the Universe - Local geometry, Shape of the Universe - Global geometry

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The root of topology was in the study of geometry in ancient cultures. Leibniz was the first to employ the term analysus situs, later employed in the 19th century to refer to what is now known as topology. Yet, this is only part of the argument in favor of his anticipation. As Benoit Mandelbrot, in his monumental The Fractal Geometry of Nature, wrote (taken from Leibniz's Cultural Pluralism And N ...

See also:

Topology, Topology - History, Topology - Elementary introduction, Topology - Some theorems in general topology, Topology - Some useful notions from algebraic topology, Topology - Outline of the deeper theory, Topology - Generalizations

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Throughout, let K stand for one of the fields R or C. The familiar Euclidean spaces Kn, where the Euclidean norm of x = (x1, ..., xn) is given by ||x|| = (∑ |xi|2)1/2, are Banach spaces. The space of all continuous functions f : [a, b] → K defined on a closed interval [a, b] becomes a Banach space if we define the norm of such a f ...

See also:

Banach space, 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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Let X be a compact space that is a subspace of Euclidean space E of dimension n. Let Y be the complement of X in E. Then if H stands for reduced homology or reduced cohomology, with coefficients in a given abelian group, there is an isomorphism between Hq(X) and Hn − q − 1(Y). ...

See also:

Alexander duality, Alexander duality - Modern statement, Alexander duality - Alexander's 1915 result

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Note the similarity between the definitions of compact and paracompact: for paracompact, we replace "subcover" by "open refinement" and "finite" by "locally finite". Both of these changes are significant: if we take the above definition of paracompact and change "open refinement" back to "subcover", or "locally finite" back to "finite", we end up with the compact spaces in both cases. A hereidtarily paracompact space is a space such that every subspace of it is paracompact. This is equivalent to requiring that every open subspace be p ...

See also:

Paracompact space, Paracompact space - Definitions of relevant terms, Paracompact space - Examples and counterexamples, Paracompact space - Properties, Paracompact space - Partitions of unity, Paracompact space - Variations, Paracompact space - Similarities with compactness, Paracompact space - Product related properties

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The cofinite topology (sometimes called the finite complement topology) is a topology which can be defined on every set X. It has precisely the empty set and all cofinite subsets of X as open sets. As a consequence, in the cofinite topology, the only closed subsets are finite sets, or the whole of X. Then X is automatically compact in this topology, since every open set only omits finitely many points of X. Also, the cofinite topology is the smallest topology satisfying the T1 axiom ...

See also:

Cofinite, Cofinite - Boolean algebras, Cofinite - Cofinite topology, Cofinite - Double-pointed cofinite topology

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Locally compact space - Compact Hausdorff spaces. Every compact Hausdorff space is also locally compact, and many examples of compact spaces may be found in the article Compact space. Here we mention only: the unit interval [0,1]; any closed topological manifold; the Cantor set; the Hilbert cube. Locally compact spa ...

See also:

Locally compact space, Locally compact space - Examples and nonexamples, Locally compact space - Compact Hausdorff spaces, Locally compact space - Locally compact Hausdorff spaces that are not compact, Locally compact space - Hausdorff spaces that are not locally compact, Locally compact space - Facts about locally compact Hausdorff spaces, Locally compact space - The point at infinity, Locally compact space - Locally compact groups, Locally compact space - Non-Hausdorff spaces

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Given a topological space (X,τ) and a subset , the subspace topology on S is defined by 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,τ). If S is equipped with the subspace topology then it is a topological space in its own right ...

See also:

Subspace topology, Subspace topology - Definition, Subspace topology - Examples, Subspace topology - Properties, Subspace topology - Preservation of topological properties

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To define a structure sheaf on Spec(R), we first define Df to be the set of all prime ideals P in Spec(R) such that f is not in P. {Df}f∈R is a basis for the topology of open sets. We define a sheaf on the Df by setting Γ(Df, OX)=Rf, the localization of R at the multiplicative system {1,f,f2,f3See also:

Spectrum of a ring, 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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The Kaluza-Klein theory is striking because it has a particularly elegant presentation in terms of geometry. In a certain sense, it looks just like ordinary gravity in free space, except that it is phrased in five dimensions instead of four. Kaluza-Klein theory - The Einstein equations. The equations governing ordinary gravity in free space can be obtained from an action, by applying the variational principle to a certain action. Let M be a (pseudo-)Riemannian manifold, which may be taken as the spa ...

See also:

Kaluza-Klein theory, Kaluza-Klein theory - Overview, Kaluza-Klein theory - Space-Time-Matter theory, Kaluza-Klein theory - Geometric interpretation, Kaluza-Klein theory - The Einstein equations, Kaluza-Klein theory - The Maxwell equations, Kaluza-Klein theory - The Kaluza-Klein geometry, Kaluza-Klein theory - Commentary and generalizations

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A metric space is a 2-tuple (X,d) where X is a set and d is a metric on X, that is, a function d : X × X → R such that d(x, y) ≥ 0     (non-negativity) d(x, y) = 0   if and only if   x = y     (identity of indiscernibles) d(x, y) = d(y, x)     ...

See also:

Metric space, Metric space - History, Metric space - Definition, Metric space - Examples, Metric space - Metric spaces as topological spaces, Metric space - Boundedness and compactness, Metric space - Separation properties and extension of continuous functions, Metric space - Distance between points and sets, Metric space - Equivalence of metric spaces, Metric space - Quotient metric space

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For any ordinal number λ one can consider the spaces of ordinal numbers together with the natural order topology. These spaces are called ordinal spaces. (Note that in the usual set-theoretic construction of ordinal numbers we have λ = [0,λ) and λ + 1 = [0,λ]). Obviously, these spaces are mostly of interest when λ is an infinite ordinal; otherwise (for finite ordinals), the order topology is simply the discrete topology. When λ = ω (the first infinite ordinal), the space [0,ω) is just N with the usual topology, while [0,ω ...

See also:

Order topology, Order topology - Induced order topology, Order topology - Ordinal space, Order topology - Left and right order topologies

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As mentioned in the previous section, any compact Hausdorff space is also locally compact, and any locally compact Hausdorff space is in fact a Tychonoff space. Every locally compact Hausdorff space is a Baire space. That is, the conclusion of the Baire category theorem holds: the interior of every union of countably many nowhere dense subsets is empty. A subspace X of a locally compact Hausdorff space Y is locally compact if and only if X can be written as the set-theoretic difference of two closed subsets ...

See also:

Locally compact space, Locally compact space - Examples and nonexamples, Locally compact space - Compact Hausdorff spaces, Locally compact space - Locally compact Hausdorff spaces that are not compact, Locally compact space - Hausdorff spaces that are not locally compact, Locally compact space - Facts about locally compact Hausdorff spaces, Locally compact space - The point at infinity, Locally compact space - Locally compact groups, Locally compact space - Non-Hausdorff spaces

Read more here: » Locally compact space: Encyclopedia II - Locally compact space - Facts about locally compact Hausdorff spaces

Several interesting variants of the order topology can be given: The left order topology on X is the topology whose open sets consist of intervals of the form (a, ∞). The right order topology on X is the topology whose open sets consist of intervals of the form (−∞, b). The left and right order topologies can be used to give counterexamples in general topology. For example, the left or right order topology on a bounded ...

See also:

Order topology, Order topology - Induced order topology, Order topology - Ordinal space, Order topology - Left and right order topologies

Read more here: » Order topology: Encyclopedia II - Order topology - Left and right order topologies