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Type of mathematical space
In mathematics, especially general topology and analysis, compactness is a property of a space that makes it behave in many ways like a finite set. For
Compact_space
Type of topological space in mathematics
topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely
Locally_compact_space
Topological space where every sequence has a convergent subsequence
In mathematics, a topological space X {\displaystyle X} is sequentially compact if every sequence of points in X {\displaystyle X} has a convergent subsequence
Sequentially_compact_space
Type of topological space
space is said to be σ-compact if it is the union of countably many compact subspaces. A space is said to be σ-locally compact if it is both σ-compact
Σ-compact_space
related branches of mathematics, a core-compact topological space X {\displaystyle X} is a topological space whose partially ordered set of open subsets
Core-compact_space
Mathematics concept
facts: Every compact space is feebly compact. Every feebly compact paracompact space is compact.[citation needed] Every feebly compact space is pseudocompact
Feebly_compact_space
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
example of a countably compact space that is not compact. Every compact space is countably compact. A countably compact space is compact if and only if it
Countably_compact_space
In topology, the core of a locally compact space is a cardinal invariant of a locally compact space X {\displaystyle X} , denoted by cor ( X ) {\displaystyle
Core of a locally compact space
Core_of_a_locally_compact_space
Generalization of compactness
ambient space). The term precompact (or pre-compact) is sometimes used with the same meaning, but precompact is also used to mean relatively compact. These
Totally_bounded_space
by the space of continuous functions on a compact Hausdorff space X {\displaystyle X} with values in the real or complex numbers. This space, denoted
Space of continuous functions on a compact space
Space_of_continuous_functions_on_a_compact_space
Topological space which is a generalization of certain compact spaces
by Dieudonné (1944). Every compact space is paracompact. Every paracompact Hausdorff space is normal, and a Hausdorff space is paracompact if and only
Paracompact_space
Type of topological space in mathematics
In mathematics, a topological space X {\displaystyle X} is said to be limit point compact or weakly countably compact if every infinite subset of X {\displaystyle
Limit_point_compact
Inputs for which a function's value is non-zero
continuous. Functions with compact support on a topological space X {\displaystyle X} are those whose closed support is a compact subset of X . {\displaystyle
Support_(mathematics)
Subset of a topological space whose closure is compact
subset of a compact topological space is relatively compact (since a closed subset of a compact space is compact). In an arbitrary topological space every subset
Relatively_compact_subspace
Functional analysis concept
compact operator on Hilbert space is an extension of the concept of a matrix acting on a finite-dimensional vector space; in Hilbert space, compact operators
Compact operator on Hilbert space
Compact_operator_on_Hilbert_space
Type of topological space
commonly used notion of compactness, which requires the existence of a finite subcover. A hereditarily Lindelöf space is a topological space of which every subspace
Lindelöf_space
On when a family of real, continuous functions has a uniformly convergent subsequence
with domain a compact metric space (Dunford & Schwartz 1958, p. 382). Modern formulations of the theorem allow for the domain to be compact Hausdorff and
Arzelà–Ascoli_theorem
Topics referred to by the same term
contain them Compact operator, a linear operator that takes bounded subsets to relatively compact subsets, in functional analysis Compact space, a topological
Compact
Embedding a topological space into a compact space as a dense subset
result of making a topological space into a compact space. A compact space is a space in which every open cover of the space contains a finite subcover.
Compactification (mathematics)
Compactification_(mathematics)
Topology on Cartesian products of topological spaces
is compact (Tychonoff's theorem). A product of locally compact spaces need not be locally compact. However, an arbitrary product of locally compact spaces
Product_topology
British musician
the Cure ended in 2005. Bamonte was one of three members in the band Compact Space, along with Christian Eigner and Florian Kraemmer. In the band, Bamonte
Daryl_Bamonte
Normed vector space that is complete
but a compact ball/neighborhood exists if and only if X {\displaystyle X} is finite-dimensional. In particular, no infinite–dimensional normed space can
Banach_space
Homology theory for locally compact spaces
homology theory for locally compact spaces, introduced by Armand Borel and John Moore in 1960. For reasonable compact spaces, Borel−Moore homology coincides
Borel–Moore_homology
Mathematical space with a notion of closeness
Quasitopological space – Function in topology Relatively compact subspace – Subset of a topological space whose closure is compact Space (mathematics) –
Topological_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
Type of continuous linear operator
infinite-dimensional spaces, bounded sets are usually not compact, and bounded sequences need not have convergent subsequences. Compact operators partly restore
Compact_operator
Way to extend a non-compact topological space
extend a noncompact topological space by adjoining a single point in such a way that the resulting space is compact. It is named after the Russian mathematician
Alexandroff_extension
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
Type of mathematical convergence in topology
It is associated with the compact-open topology. Let ( X , T ) {\displaystyle (X,{\mathcal {T}})} be a topological space and ( Y , d Y ) {\displaystyle
Compact_convergence
Statement about linear functionals and measures
representation theorem relates linear functionals on spaces of continuous functions on a locally compact space to measures in measure theory. The theorem is
Riesz–Markov–Kakutani representation theorem
Riesz–Markov–Kakutani_representation_theorem
Vector space with a notion of nearness
Normed space – Vector space on which a distance is definedPages displaying short descriptions of redirect targets Locally compact field Locally compact group –
Topological_vector_space
Curve whose range contains the unit square
unit square. (Alternatively, we could use the theorem that every compact metric space is a continuous image of the Cantor set to get the function f {\displaystyle
Space-filling_curve
Product of any collection of compact topological spaces is compact
Tychonoff's theorem states that the product of any collection of compact topological spaces is compact with respect to the product topology. The theorem is named
Tychonoff's_theorem
Concept in measure theory
tight if and only if it is sequentially weakly compact. If X {\displaystyle X} is a metrizable compact space, then every collection of (possibly complex)
Tightness_of_measures
Branch of topology
a compact space is compact. A compact subset of a Hausdorff space is closed. Every continuous bijection from a compact space to a Hausdorff space is
General_topology
Relation among continuous functions
which states that a subset of C(X), the space of continuous functions on a compact Hausdorff space X, is compact if and only if it is closed, pointwise
Equicontinuity
In mathematics, an ω-bounded space is a topological space in which the closure of every countable subset is compact. More generally, if P is some property
Ω-bounded_space
Type of topological space
locally compact at a point x ∈ X if x lies in the interior of some compact subset of X. X is a locally compact space if it is locally compact at every
Polyadic_space
Type of mathematical measure
the σ-algebra of Borel sets of a Hausdorff topological space X that is finite on all compact sets, outer regular on all Borel sets, and inner regular
Radon_measure
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
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
Eberlein, is a compact topological space homeomorphic to a subset of a Banach space with the weak topology. Every compact metric space, more generally
Eberlein_compactum
Topics referred to by the same term
Weakly compact set, a compact set in a space with the weak topology Weakly compact set, a set that has some but not all of the properties of compact sets
Weakly_compact
Hausdorff space or weakly Hausdorff space is a topological space where the image of every continuous map from a compact Hausdorff space into the space is closed
Weak_Hausdorff_space
Mathematical theorem in the study of analysis
functions on a compact Hausdorff space. Further, there is a generalization of the Stone–Weierstrass theorem to noncompact Tychonoff spaces, namely, any
Stone–Weierstrass_theorem
Continuous real function on a closed interval has a maximum and a minimum
also true for an upper semicontinuous function. (see compact space#Functions and compact spaces). We look at the proof for the upper bound and the maximum
Extreme_value_theorem
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
Type of topological group in mathematics
K(LCA). Compact group – Topological group with compact topology Complete field Locally compact field Locally compact space – Type of topological space in mathematics
Locally_compact_group
Digital optical disc data storage format
The compact disc (CD) is a digital optical disc data storage format co-developed by Philips and Sony to store and play digital audio recordings. It employs
Compact_disc
Concept in mathematical topology
topological space is said to be hemicompact if it has a sequence of compact subsets such that every compact subset of the space lies inside some compact set in
Hemicompact_space
subcover. Every compact space is Lindelöf and paracompact. Therefore, every compact Hausdorff space is normal. See also quasicompact. Compact-open topology
Glossary_of_general_topology
Concept in topology
In mathematics, a maximal compact subgroup K of a topological group G is a subgroup K that is a compact space, in the subspace topology, and maximal amongst
Maximal_compact_subgroup
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
pseudocompact (see Engelking, p. 153). Compact space Paracompact space Normal space Pseudocompact space Tychonoff space Gillman, Leonard; Jerison, Meyer, "Rings
Realcompact_space
field of general topology, a topological space is said to be mesocompact if every open cover has a compact-finite open refinement. That is, given any
Mesocompact_space
(pseudo-)Riemannian manifold whose geodesics are reversible
than the Riemannian definition, and reduces to it when H is compact. Riemannian symmetric spaces arise in a wide variety of situations in both mathematics
Symmetric_space
Topological concept
compact space is finite. Indeed, let G = { G a | a ∈ A } {\displaystyle G=\{G_{a}|a\in A\}} be a locally finite family of subsets of a compact space X
Locally_finite_collection
Mathematical set with some added structure
projective space is also a topological space. An affine space is a non-compact manifold; a projective space is a compact manifold. In a real projective space a
Space_(mathematics)
Lemma in topology
many compact spaces is compact. The lemma uses the following terminology: If X {\displaystyle X} and Y {\displaystyle Y} are topological spaces and X
Tube_lemma
space X is cocompact if the quotient space X/G is a compact space. If X is locally compact, then an equivalent condition is that there is a compact subset
Cocompact_group_action
Type of topology
mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. The compact-open topology is
Compact-open_topology
Vehicle size class
Compact MPV (an abbreviation for Compact Multi-Purpose Vehicle) is a vehicle size class for the middle size of MPVs. The Compact MPV size class sits between
Compact_MPV
compactness, since a compact subset of a Hausdorff space is closed. Thus, every compact Hausdorff space is H-closed. The notion of an H-closed space has
H-closed_space
Sufficient criterion for uniform convergence
monotone sequence of continuous functions converges pointwise on a compact space and if the limit function is also continuous, then the convergence is
Dini's_theorem
Tool in algebraic topology
→ Hj(X,E), which is an isomorphism for X compact. For a sheaf E on a locally compact space X, the compactly supported cohomology of X × R with coefficients
Sheaf_cohomology
to functions defined on normed vector spaces and the other applying to functions defined on locally compact spaces. Aside from this difference, both of
Vanish_at_infinity
Algebraic ring without a multiplicative identity
decreasing to zero at infinity, especially those with compact support on some (non-compact) space. Rngs appear in the following chain of class inclusions:
Rng_(algebra)
Vector bundle existing over a Grassmannian
vector bundle (over a compact space) is a pullback of the tautological bundle; this is to say a Grassmannian is a classifying space for vector bundles.
Tautological_bundle
Concept in topology
products and disjoint unions of countable families of Polish spaces, locally compact spaces that are metrizable and countable at infinity, countable intersections
Polish_space
Particular kind of algebraic structure
{\displaystyle C_{0}(X)} , the space of (complex-valued) continuous functions, defined on a locally compact Hausdorff space X {\displaystyle X} , that vanish
Banach_algebra
Index of articles associated with the same name
dense subset Lindelöf space: every open cover has a countable subcover σ-compact space: there exists a countable cover by compact spaces These axioms are related
Axiom_of_countability
Mathematical property of a space
Lindelöf. A space is Lindelöf if every open cover has a countable subcover. σ-compact. A space is σ-compact if it is the union of countably many compact subspaces
Topological_property
Number representing a continuous quantity
connected), separable and complete metric space of Hausdorff dimension 1. The real numbers are locally compact but not compact. There are various properties that
Real_number
Topological space with a point-finite open refinement for every cover
metacompact space is orthocompact. Every metacompact normal space is a shrinking space The product of a compact space and a metacompact space is metacompact
Metacompact_space
Topological space with a bounded image under any continuous function to R
easily proved: the image of a compact space under any continuous function is compact, and every compact set in a metric space is bounded. If Y is the continuous
Pseudocompact_space
Topics referred to by the same term
Compactness can refer to: Compact space, in topology Compact operator, in functional analysis Compactness theorem, in first-order logic Compactness measure
Compactness_(disambiguation)
and analysis, an exhaustion by compact sets of a topological space X {\displaystyle X} is a nested sequence of compact subsets K i {\displaystyle K_{i}}
Exhaustion_by_compact_sets
Class of mathematical sets
all Hausdorff σ-compact spaces, but can be different in more pathological spaces. In the case that X {\displaystyle X} is a metric space, the Borel algebra
Borel_set
Bounded sequence in finite-dimensional Euclidean space has a convergent subsequence
\mathbb {R} ^{n}} is compact if and only if it is closed and bounded. In fact, general topology tells us that a metrizable space is compact if and only if it
Bolzano–Weierstrass_theorem
Banach space stating that all weakly compact operators from this space into another Banach space are completely continuous. Many standard Banach spaces have
Dunford–Pettis_property
Theorem in functional analysis
theorem) states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology. A common proof identifies the unit
Banach–Alaoglu_theorem
Fraction of a space filled by objects packed into that space
{\displaystyle K_{1},\dots ,K_{n}} are measurable subsets of a compact measure space X {\displaystyle X} and their interiors pairwise do not intersect
Packing_density
Topological space with a dense countable subset
cardinality of a topological space: any set endowed with the trivial topology is separable, as well as second countable, quasi-compact, and connected. The "trouble"
Separable_space
Line formed by the real numbers
differentiable structure that the topological space supports.) The real line is a locally compact space and a paracompact space, as well as second-countable and normal
Number_line
Mode of convergence of a function sequence
uniformly convergent sequence is compactly convergent. For locally compact spaces local uniform convergence and compact convergence coincide. A sequence
Uniform_convergence
Second-countable space Separable space Lindelöf space Sigma-compact space Connected space Simply connected space Path connected space T0 space T1 space Hausdorff
List of general topology topics
List_of_general_topology_topics
Mathematical map between topological spaces
mathematics, a function between topological spaces is called proper if inverse images of compact subsets are compact. In algebraic geometry, the analogous concept
Proper_map
considered the arbitrary product of topological spaces and who proved in the 1930s that the Tychonoff cube is compact. Tychonoff later generalized this to the
Tychonoff_cube
Concept in mathematics
line). The configuration space Conf n ( X ) {\displaystyle \operatorname {Conf} _{n}(X)} of distinct points is non-compact, having ends where the points
Configuration space (mathematics)
Configuration_space_(mathematics)
Type of topological space
of these statements are: every locally compact Hausdorff space is Tychonoff, and every compact Hausdorff space is normal Hausdorff. The following results
Hausdorff_space
Duality for locally compact abelian groups
finite-dimensional vector space over the reals or a p-adic field. The Pontryagin dual of a locally compact abelian group is the locally compact abelian topological
Pontryagin_duality
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 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
Theorem about the intersections of d-dimensional convex sets
finite intersection property characterization of compactness: a collection of closed subsets of a compact space has a non-empty intersection if and only if
Helly's_theorem
Electrical connector used in stagecraft
principal advantages of powerCON include its high current capacity in a compact space – it is smaller than an IEC connector yet provides double the current-carrying
PowerCon
{\displaystyle {\mathcal {C}}(X,\mathbb {K} )} , where X {\displaystyle X} is a compact space and K {\displaystyle \mathbb {K} } either the real numbers or the complex
Haar_space
Duality for sheaves of k-modules over a locally compact space
in 1965 by Jean-Louis Verdier (1965) as an analog for locally compact topological spaces of Alexander Grothendieck's theory of Poincaré duality in étale
Verdier_duality
On topological spaces where the intersection of countably many dense open sets is dense
topological space is a Baire space. More generally, every complete pseudometric space is a Baire space. (BCT2) Every locally compact Hausdorff space is a Baire
Baire_category_theorem
Topological group with compact topology
mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space. Compact groups are a natural
Compact_group
COMPACT SPACE
COMPACT SPACE
Girl/Female
Hindu, Indian
Compare
Girl/Female
Tamil
Compare
Surname or Lastname
Americanized spelling of German Kahle. Compare Kahley or Köhler (see Kohler).English and Manx
Americanized spelling of German Kahle. Compare Kahley or Köhler (see Kohler).English and Manx : variant spelling of Caley.
Boy/Male
Indian, Tamil
No Compare
Boy/Male
Indian, Punjabi, Sikh
Company of Guru
Girl/Female
Indian, Telugu
Good Company
Boy/Male
Indian, Sanskrit
Fallen from Glory
Boy/Male
Hindu, Indian
Compact; Firm; Solid
Girl/Female
Arabic
Sensible Contact
Girl/Female
Muslim
Beauty of company
Boy/Male
Hindu, Indian, Sanskrit
Company
Boy/Male
Indian, Punjabi, Sikh
Lord's Company
Girl/Female
Arabic, Muslim
Beauty of Company
Surname or Lastname
Americanized form of German Eisele. Compare Isley.English
Americanized form of German Eisele. Compare Isley.English : unexplained. This name is quite widespread in Britain.
Boy/Male
Indian, Punjabi, Sikh
Liberation through Company
Girl/Female
Hindu, Indian, Marathi, Tamil
Compact; Promise
Boy/Male
Hindu, Indian
Compact; Safe; Secure
Girl/Female
Arabic, Muslim
Beauty of Company
Boy/Male
Hindu, Indian, Sanskrit
In the Company
Boy/Male
Indian, Punjabi, Sikh
Good Company
COMPACT SPACE
COMPACT SPACE
Girl/Female
Tamil
Sangrama | ஸஂகà¯à®°à®®à®¾à®‚
Name of a Raga
Girl/Female
Hindu
Surname or Lastname
English
English : habitational name from a place in Cheshire named Kelsall, from the Middle English personal name Kell + Old English halh ‘nook or corner of land’, or possibly from Kelshall in Hertfordshire, which is named with an Old English personal name Cylli + Old English hyll ‘hill’, or even Kelsale in Suffolk, named with an Old English personal name Cēl(i) or Cēol + Old English halh.
Boy/Male
Australian, Chinese, French
Great; Vast; Morning
Boy/Male
Indian
Aggressive, Hardliner
Boy/Male
American, Anglo, Australian, British, English, German, Norse, Norwegian, Scandinavian, Swedish
Proud; Firebrand; Sword Blade; Sword; Fiery Torch; Beacon
Girl/Female
Hebrew
Tender.
Girl/Female
Biblical
Rebellious people.
Boy/Male
Indian, Punjabi, Sikh
God's Support
Boy/Male
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Lord Krishna
COMPACT SPACE
COMPACT SPACE
COMPACT SPACE
COMPACT SPACE
COMPACT SPACE
adv.
In a compact manner; with close union of parts; densely; tersely.
p. p. & a
Brief; close; pithy; not diffuse; not verbose; as, a compact discourse.
n.
An association of persons for the purpose of carrying on some enterprise or business; a corporation; a firm; as, the East India Company; an insurance company; a joint-stock company.
n.
The crew of a ship, including the officers; as, a whole ship's company.
p. pr. & vb. n.
of Compact
n.
Contact or impression by touch; collision; forcible contact; force communicated.
a.
Strong; firm; compact.
n.
An inclosing limit; boundary; circumference; as, within the compass of an encircling wall.
v. t.
To compact or join anew.
a.
Compact; pressed close; concentrated; firmly united.
n.
A mixture for fertilizing land; esp., a composition of various substances (as muck, mold, lime, and stable manure) thoroughly mingled and decomposed, as in a compost heap.
v. i.
To be like or equal; to admit, or be worthy of, comparison; as, his later work does not compare with his earlier.
v. i.
To bear or endure; to put up (with); as, to comport with an injury.
n.
One who makes a compact.
v. t.
To manure with compost.
n.
Extent; reach; sweep; capacity; sphere; as, the compass of his eye; the compass of imagination.
n.
Guests or visitors, in distinction from the members of a family; as, to invite company to dine.
v. t.
To mingle, as different fertilizing substances, in a mass where they will decompose and form into a compost.
imp. & p. p.
of Compact