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Property of topological spaces
a topological space X {\displaystyle X} is called a compactly generated space or k-space if its topology is determined by compact spaces in a manner made
Compactly_generated_space
Type of mathematical space
cube is compact, again a consequence of Tychonoff's theorem. A profinite group (e.g. Galois group) is compact. Compactly generated space Compactness theorem
Compact_space
mathematics, a compactly generated (topological) group is a topological group G which is algebraically generated by one of its compact subsets. This should
Compactly_generated_group
Topics referred to by the same term
mathematics, compactly generated can refer to: Compactly generated group, a topological group which is algebraically generated by one of its compact subsets
Compactly_generated
-Hausdorff spaces are to Δ {\displaystyle \Delta } -generated spaces as weak Hausdorff spaces are to compactly generated spaces. Fixed-point space – Space where
Weak_Hausdorff_space
Type of topology
coincides with the compact-open topology and may be used to uniquely define it. The modification of the definition for compactly generated spaces may be viewed
Compact-open_topology
Type of topological space in mathematics
compact. Quotient spaces of locally compact Hausdorff spaces are compactly generated. Conversely, every compactly generated Hausdorff space is a quotient
Locally_compact_space
Category used in algebraic topology
of compactly generated weak Hausdorff spaces, CGWH, is a category used in algebraic topology as an alternative to the category of topological spaces, Top
Category of compactly generated weak Hausdorff spaces
Category_of_compactly_generated_weak_Hausdorff_spaces
Topics referred to by the same term
for a compactly generated space in topology K-space (functional analysis) is an F-space such that every twisted sum by the real line splits K-Space (band)
K-space
Category whose objects are topological spaces and whose morphisms are continuous maps
with compactly generated spaces as objects and continuous maps as morphisms or with the category of compactly generated weak Hausdorff spaces. Like many
Category of topological spaces
Category_of_topological_spaces
Continuous deformation between two continuous functions
difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra. Formally, a homotopy
Homotopy
Type of mathematical convergence in topology
{\displaystyle (X,{\mathcal {T}})} is a compactly generated space, f n → f {\displaystyle f_{n}\to f} compactly, and each f n {\displaystyle f_{n}} is
Compact_convergence
Topological space characterized by sequences
"convenient". Every sequential space is compactly generated, and finite products in Seq coincide with those for compactly generated spaces, since products in the
Sequential_space
Finest topology making some functions continuous
here. As an example, a commonly used variant of the notion of compactly generated space is defined as the final topology with respect to a proper class
Final_topology
of non-compactness Paracompact space Locally compact space Compactly generated space Axiom of countability Sequential space First-countable space Second-countable
List of general topology topics
List_of_general_topology_topics
Algebraic structure of set algebra
containing all the others. It is the algebra generated by the product topology or weak topology on the product space { H , T } ∞ . {\displaystyle \{H,T\}^{\infty
Σ-algebra
Relation among continuous functions
the change in functions More generally, on any compactly generated space; e.g., a first-countable space. Rudin 1991, p. 44 §2.5. Reed & Simon (1980), p
Equicontinuity
Mathematical concept
D({\text{Sh}}(X;{\text{Ab}}))} for a non-compact topological space X {\displaystyle X} , it is generally not a compactly generated category. Some evidence for this
Compact_object_(mathematics)
Banach space ℓ 1 {\displaystyle \ell ^{1}} is not a K-space. Compactly generated space – Property of topological spaces Gelfand–Shilov space Kalton,
K-space_(functional_analysis)
Type of topological space
that X and Y are compactly generated Hausdorff spaces, so Hom(X,Y) is often taken with the compactly generated variant of the compact-open topology; the
CW_complex
Concept in homotopy theory
{\displaystyle g\colon A\to B} is any (continuous) map (between compactly generated spaces), and i : A → X {\displaystyle i\colon A\to X} is a cofibration
Cofibration
On when a family of real, continuous functions has a uniformly convergent subsequence
be locally compact or compactly generated, and the functions can assume values in a metric space or (Hausdorff) uniform space with only minimal changes
Arzelà–Ascoli_theorem
Topological space where each point has a countable neighbourhood basis
first-countable space is compactly generated. Every subspace of a first-countable space is first-countable. Any countable product of a first-countable space is first-countable
First-countable_space
Moreover, a (Hausdorff) space is realcompact if and only if it has the uniform topology and is complete for the uniform structure generated by the continuous
Realcompact_space
Type of topological space in mathematics
topology, i.e., the topology generated by all intervals ( x , ∞ ) . {\displaystyle (x,\infty ).} The space is limit point compact because given any point a
Limit_point_compact
Tiling of hyperbolic 3-space by uniform polyhedra
hyperbolic space is a uniform tessellation of uniform polyhedral cells. In 3-dimensional hyperbolic space there are nine Coxeter group families of compact convex
Uniform honeycombs in hyperbolic space
Uniform_honeycombs_in_hyperbolic_space
Topology determined by family of subspaces
subspaces. Compactly generated spaces (in the sense of Definition 1 in that article) are those determined by the family of all their compact subspaces
Coherent_topology
Category admitting tensor products
product and R as the unit. The category of pointed spaces (restricted to compactly generated spaces for example) is monoidal with the smash product serving
Monoidal_category
Wavelet constructed using a spline function
j}c_{m+n-j-1}.} The spline wavelets generated using the interpolatory wavelets are not compactly supported. Compactly supported B-spline wavelets were discovered
Spline_wavelet
Combination of pointed topological spaces
arguments). For any pointed spaces X, Y, and Z in an appropriate "convenient" category (e.g., that of compactly generated spaces), there are natural (basepoint
Smash_product
Topological concept in mathematics
non-compact, but this is not an open manifold since the circle (one of its components) is compact. Most books generally define a manifold as a space that
Closed_manifold
Type of topology in mathematics
preorder on the space. Spaces with an Alexandrov topology are also known as Alexandrov-discrete spaces or finitely generated spaces. The latter name
Alexandrov_topology
Statement about linear functionals and measures
Cc(X), the space of compactly supported complex-valued continuous functions, is as follows: Theorem Let X be a locally compact Hausdorff space and ψ {\displaystyle
Riesz–Markov–Kakutani representation theorem
Riesz–Markov–Kakutani_representation_theorem
Type of mathematical measure
theory for locally compact spaces from the functional-analytic viewpoint, it is necessary to extend measure (integral) from compactly supported continuous
Radon_measure
Mathematical space with a notion of closeness
For any indexed family of topological spaces, the product can be given the product topology, which is generated by the inverse images of open sets of
Topological_space
Theorem in mathematical logic
compactness theorem for the propositional calculus is a consequence of Tychonoff's theorem (which says that the product of compact spaces is compact)
Compactness_theorem
Category in mathematics
explain the name: if a localizing subcategory S of a compactly generated triangulated category D is generated by a set of objects, then there is a Bousfield
Triangulated_category
Vector space with a notion of nearness
Locally compact quantum group Locally convex topological vector space – Space with topology generated by convex sets Ordered topological vector space Topological
Topological_vector_space
widely used, the Baire sets of a locally compact Hausdorff space form the smallest σ-algebra such that all compactly supported continuous functions are measurable
Baire_set
mathematics a Brauner space is a complete compactly generated locally convex space X {\displaystyle X} having a sequence of compact sets K n {\displaystyle
Brauner_space
Generalized function whose value is zero everywhere except at zero
a compactly supported function, are needed to ensure pointwise convergence almost everywhere. If the initial η = η1 is itself smooth and compactly supported
Dirac_delta_function
be equipped with the direct limit topology, making it again a compactly generated space. One can then identify SP(X) respectively Z[X] with B(N, X) respectively
Symmetric_product_(topology)
Class of mathematical sets
be generated by the compact sets of the topological space, rather than the open sets. The two definitions are equivalent for many well-behaved spaces, including
Borel_set
Categorical treatment of topological spaces
category is a category that is enriched over the category of compactly generated Hausdorff spaces. They can be used as a foundation for higher category theory
Topological category (enriched category theory)
Topological_category_(enriched_category_theory)
Subset of Euclidean space is compact if and only if it is closed and bounded
S} of Euclidean space R n {\displaystyle \mathbb {R} ^{n}} , the following two statements are equivalent: S {\displaystyle S} is compact, that is, every
Heine–Borel_theorem
Normed vector space that is complete
vector space – Space with topology generated by convex sets Modulus and characteristic of convexity Smith space Topological vector space – Vector space with
Banach_space
Tool in algebraic topology
finitely generated R-module. Then the cohomology groups Hj(X,E) are finitely generated R-modules. For example, for a compact Hausdorff space X that is
Sheaf_cohomology
Space with topology generated by convex sets
vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with
Locally convex topological vector space
Locally_convex_topological_vector_space
Type of vector space in math
with domain the Sobolev space H1(R). Equivalently, it is the closure of this differential operator initially defined on compactly supported smooth functions
Hilbert_space
Manifold with inversion symmetry
space, a homogeneous space for SU(2) and SL(2,C). Irreducible compact Hermitian symmetric spaces are exactly the homogeneous spaces of simple compact
Hermitian_symmetric_space
Branch of topology
larger space with the subset. For any indexed family of topological spaces, the product can be given the product topology, which is generated by the inverse
General_topology
Group type in algebra
generated. Every quotient of a finitely generated group G is finitely generated; the quotient group is generated by the images of the generators of G under
Finitely_generated_group
locally compact abelian group G {\displaystyle G} , let A {\displaystyle A} be a compactly generated subgroup, and B {\displaystyle B} a compact subgroup
Schwartz–Bruhat_function
The compact elements of Sub(A) are exactly the finitely generated substructures of A. Every substructure is the union of its finitely generated substructures;
Compact_element
Mathematical map between topological spaces
{\displaystyle Y} is a compactly generated Hausdorff space (this includes Hausdorff spaces that are either first-countable or locally compact), then f {\displaystyle
Proper_map
Smith space is a complete compactly generated locally convex topological vector space X {\displaystyle X} having a universal compact set, i.e. a compact set
Smith_space
Concept in topology
"most general" compact Hausdorff space generated by X, in the sense that any continuous map from X into any other compact Hausdorff space factors uniquely
Stone–Čech_compactification
American spaceflight and AI company
the company. Starlink has generated the bulk of SpaceX's income and led to its Starshield military counterpart. In 2020, SpaceX began to operate its Dragon
SpaceX
algebra is the C*-algebra generated by the unilateral shift on the Hilbert space l2(N). Identifying l2(N) with the Hardy space H2, the Toeplitz algebra
Toeplitz_algebra
Digital optical disc data storage format
represented generated 12% each of the French music industry revenues. Sony and Philips received praise for the development of the compact disc from professional
Compact_disc
compact subset of Y . {\displaystyle Y.} for every weakly compactly generated Banach space Y , {\displaystyle Y,} every bounded linear operator from X
Grothendieck_space
Topological space with a distinguished point
categories of spaces, such as compactly generated weak Hausdorff ones. The reduced suspension Σ X {\displaystyle \Sigma X} of a pointed space X {\displaystyle
Pointed_space
Left-invariant (or right-invariant) measure on locally compact topological group
groups that are locally compact but not compact this construction does not give Haar measure as the mean value of compactly supported functions is zero
Haar_measure
Mathematical theorem in the study of analysis
taking the real parts of the generated complex unital (selfadjoint) algebra agrees with the generated real unital algebra generated. As in the real case, an
Stone–Weierstrass_theorem
Collection of open sets used to define a topology
generated by Γ {\displaystyle \Gamma } , which is the Euclidean topology on R {\displaystyle \mathbb {R} } , is coarser than the topology generated by
Base_(topology)
Product of any collection of compact topological spaces is compact
lemma, a space is compact if and only if each ultrafilter on the space converges. With this in hand, the proof becomes easy: the (filter generated by the)
Tychonoff's_theorem
Type of topological space
Stone space, also known as a profinite space, profinite set, or Boolean space, is a compact Hausdorff totally disconnected space. Stone spaces are named
Stone_space
Design method of discrete wavelet transforms
element. In the case of one continuous (or at least with bounded variation) compactly supported scaling function with orthogonal shifts, one may make a number
Multiresolution_analysis
On representability of a contravariant functor on the category of connected CW complexes
adjoint functor. Namely, if C and D are triangulated categories with C compactly generated and F a triangulated functor commuting with arbitrary direct sums
Brown's representability theorem
Brown's_representability_theorem
Space-filling curve
The Hilbert curve (also known as the Hilbert space-filling curve) is a continuous fractal space-filling curve first described by the German mathematician
Hilbert_curve
one-for-one with the ends of topological spaces defined from the graph (Diestel & Kühn 2003). The ends of a finitely generated group are defined to be the ends
End_(topology)
Mathematical construction used in homotopy theory
actual topological spaces, there is a geometric realization functor which turns simplicial sets into compactly generated Hausdorff spaces. Most classical
Simplicial_set
Certain topology in mathematics
many members of the space. Though the subspace topology of Y = {−1} ∪ {1/n}n∈N in the section above is shown not to be generated by the induced order
Order_topology
Sum of directed areas in exterior algebra
1/2B and rotations are generated in the same way: v ′ = R v R − 1 . {\displaystyle v'=RvR^{-1}.} The rotations generated are more complex though. They
Bivector
Locally compact topological group with an invariant averaging operation
it is sufficient to ask that, for every continuous positive-definite compactly supported function f on G, the function Δ–1⁄2f has non-negative integral
Amenable_group
Amenable group Capable group Commensurability (group theory) Compact group Compactly generated group Complete group Complex reflection group Congruence subgroup
List_of_group_theory_topics
Conversion of character sequences into token sequences in computer science
into template code for compiling and executing). Regular expressions compactly represent patterns that the characters in lexemes might follow. For example
Lexical_analysis
Categorical generalization of a function space in set theory
for example, given by the full subcategory spanned by the compactly generated Hausdorff spaces. In functional programming languages, the morphism eval {\displaystyle
Exponential_object
Mathematical space with a notion of distance
defined for metric spaces. Other notions, such as continuity, compactness, and open and closed sets can be defined for metric spaces, but also in the even
Metric_space
Mathematical set with some added structure
Bergman space Berkovich space Besov space Borel space Calabi-Yau space Cantor space Cauchy space Cellular space Chu space Closure space Conformal space Complex
Space_(mathematics)
NASA/ESA space telescope launched in 1990
The Hubble Space Telescope (HST or Hubble) is a space telescope that was launched into low Earth orbit in 1990 and remains in operation. It was not the
Hubble_Space_Telescope
Concept in topology
topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable
Polish_space
Property in general topology
Corollary—Every perfect, locally compact Hausdorff space is uncountable. Proof If X {\displaystyle X} is also compact then the theorem immediately implies
Finite_intersection_property
Type of mathematical space
the map is the ideal generated by the images of primitive elements under the edge homomorphisms, which is also the ideal generated by positive-degree elements
Generalized_flag_variety
Topological vector spaces
{\displaystyle 1\leq p\leq \infty } , the vector space L c p ( U ) {\displaystyle L_{\text{c}}^{p}(U)} of compactly supported L p {\displaystyle L^{p}} functions
Spaces of test functions and distributions
Spaces_of_test_functions_and_distributions
Doughnut-shaped surface of revolution
: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanar
Torus
Embedding of data within a manifold based on a similarity function
the data into a compact latent representation. VAEs are known for their ability to generate new data samples from the learned latent space. Multimodality
Latent_space
2019 video game
Space Engineers is a voxel-based sandbox game, developed and published by Czech independent developer Keen Software House. In 2013, the initial developmental
Space_Engineers
Abstract mathematical spacetime
{x}{t}}\right)} Misner space is a standard example for the study of causality since it contains both closed timelike curves and a compactly generated Cauchy horizon
Misner_space
1983 video game
Space Ace is a LaserDisc video game produced by Bluth Group, Cinematronics and Advanced Microcomputer Systems (later renamed RDI Video Systems). It was
Space_Ace
European space mission to measure gravitational waves
The Laser Interferometer Space Antenna (LISA) is a planned European space mission to detect and measure gravitational waves—slight ripples in the fabric
Laser Interferometer Space Antenna
Laser_Interferometer_Space_Antenna
Mathematical generalization of boundedness
{B}}.} The bornology generated by the set of all topological closures of sets in B {\displaystyle {\mathcal {B}}} (that is, generated by { cl B : B ∈ B
Bornology
Type of continuous map in topology
projection is a map between topological spaces that, intuitively, locally acts like a projection of multiple copies of a space onto itself. In particular, coverings
Covering_space
Mathematical concept
be confused with the dual A' of the Banach space A). In particular, suppose X is a compact Hausdorff space. Then there is a natural homeomorphism I :
Spectrum_of_a_C*-algebra
& Henriksen (1956), is a completely regular Hausdorff space for which every finitely generated ideal of the ring of real-valued continuous functions is
Sub-Stonean_space
Branch of mathematics
is to extend the duality between spaces and algebras of functions. If X {\displaystyle X} is a compact Hausdorff space, the commutative C*-algebra C (
Noncommutative_geometry
Topology on Cartesian products of topological spaces
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
the article, spaces are objects of the category of "reasonable" spaces; e.g., the category of compactly generated weak Hausdorff spaces. Davis & Kirk
Path_space_fibration
Geometric space with six dimensions
six-dimensional Euclidean space, R 6 {\displaystyle \mathbb {R} ^{6}} , is generated by considering all real 6-tuples as 6-vectors in this space. As such it has
Six-dimensional_space
State of matter
Most artificial plasmas are generated by the application of electric and/or magnetic fields through a gas. Plasma generated in a laboratory setting and
Plasma_(physics)
COMPACTLY GENERATED-SPACE
COMPACTLY GENERATED-SPACE
Biblical
penetrated
Boy/Male
Indian, Telugu
Generated
Girl/Female
Tamil
Aninditha | அநிஂதிதா
Beautiful, Virtuous, Venerated
Aninditha | அநிஂதிதா
Girl/Female
Indian
Beautiful, Virtuous, Venerated
Boy/Male
Muslim
Honored, Venerated
Girl/Female
Tamil
Generates harmony in dance and music
Girl/Female
Indian
Beautiful, Virtuous, Venerated
Boy/Male
Hindu, Indian
Compact; Firm; Solid
Girl/Female
Tamil
Anindita | அநிஂதிதா
Beautiful, Virtuous, Venerated
Anindita | அநிஂதிதா
Girl/Female
Biblical
Penetrated.
Boy/Male
Indian
Honored, Venerated
Boy/Male
Arabic, Muslim, Sindhi
Venerated; Honoured
Girl/Female
Indian
Who is to be Venerated and Respected
Boy/Male
Hindu, Indian
Self Generated; Lord Shiva
Boy/Male
Indian, Sanskrit
Devoted; Venerated
Boy/Male
Hindu, Indian
Compact; Safe; Secure
Girl/Female
Assamese, Gujarati, Hindu, Indian, Kannada, Sindhi, Tamil, Telugu
Virtuous; Venerated
Boy/Male
Hindu
Compact, Massive, Widespread, Great, Large, Mighty, Powerful, Bright, Clear, Name of Vishnu
Girl/Female
Hindu, Indian, Marathi, Tamil
Compact; Promise
Girl/Female
Hindu
Generates harmony in dance and music
COMPACTLY GENERATED-SPACE
COMPACTLY GENERATED-SPACE
Boy/Male
Sikh
Girl/Female
Arabic, Egyptian
Intelligent
Boy/Male
Indian
Limitless
Boy/Male
French, Indian
Reborn
Boy/Male
American, Australian, Biblical, British, Chinese, Christian, English, Greek, Hebrew, Welsh
Praise; Heart; Courageous and Praise; Father; Poet; Courageous; Large Hearted
Girl/Female
Hindu, Indian, Marathi
Destiny
Girl/Female
Tamil
Boat
Boy/Male
Celtic Welsh
Beloved.
Boy/Male
Hindu, Indian, Sanskrit
Whose Body is Dharma; Lord Visnu
Girl/Female
Italian
Named for John the Baptist.
COMPACTLY GENERATED-SPACE
COMPACTLY GENERATED-SPACE
COMPACTLY GENERATED-SPACE
COMPACTLY GENERATED-SPACE
COMPACTLY GENERATED-SPACE
v. t.
To beget; to procreate; to propagate; to produce (a being similar to the parent); to engender; as, every animal generates its own species.
adv.
In a compact manner; with close union of parts; densely; tersely.
a.
Relating to autogenesis; self-generated.
imp. & p. p.
of Compact
n.
One who makes a compact.
n.
Capability of being generated.
a.
First formed or generated; original; primigenial.
a.
Not begot; not yet generated; also, having never been generated; self-existent; eternal.
a.
Capable of being generated or produced.
a.
Compact; pressed close; concentrated; firmly united.
v. t.
To regard with reverential respect; to honor with mingled respect and awe; to reverence; to revere; as, we venerate parents and elders.
adv.
In a compact manner.
imp. & p. p.
of Generate
n.
That which generates.
imp. & p. p.
of Venerate
n.
One who, or that which, generates, begets, causes, or produces.
a.
Generated by water.
p. pr. & vb. n.
of Generate
p. p. & a
Brief; close; pithy; not diffuse; not verbose; as, a compact discourse.
a.
Self-generated; produced independently.