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Type of mathematical space
especially general topology and mathematical 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
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
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
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
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
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
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
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
Inputs for which a function's value is non-zero
via convolution. In a locally compact Hausdorff space, continuous functions with compact support are dense in the space of continuous functions that vanish
Support_(mathematics)
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
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)
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
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
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
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
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
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
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
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
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
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
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
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
Application of set theory concept
metric spaces, and more generally in sequential spaces, basic topological notions such as open set, closed set, convergence, continuity, compactness and
Filters_in_topology
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
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
pseudocompact (see Engelking, p. 153). Compact space Paracompact space Normal space Pseudocompact space Tychonoff space Gillman, Leonard; Jerison, Meyer, "Rings
Realcompact_space
Homology theory for locally compact spaces
homology or homology with closed support is a homology theory for locally compact spaces, introduced by Armand Borel and John Moore in 1960. The applications
Borel–Moore_homology
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
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
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
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
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
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
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
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
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
(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
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
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
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
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)
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
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
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
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
Topological space with a bounded image under any continuous function to R
space is pseudocompact. The converse is true for metric spaces. As sequential compactness is an equivalent condition to compactness for metric spaces
Pseudocompact_space
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
Menger space is a topological space that satisfies a certain basic selection principle that generalizes σ-compactness. A Menger space is a space in which
Menger_space
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 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
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
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
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
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
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
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
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
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
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
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
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
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)
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
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)
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
Property of topological spaces stronger than normality
topology is compact, hence paracompact, and T1, but is not even normal. Every normal countably compact space (hence every normal compact space) is collectionwise
Collectionwise_normal_space
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
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
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
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
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
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
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
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
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
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
{\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
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
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
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
Science fiction novel series by C. J. Cherryh
pidgin used by Compact spacers—although they are quite eloquent in their own numerous languages. Mahendo'sat are the "glue" of the Compact, always trying
The_Chanur_novels
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
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
compact subset of Y . {\displaystyle Y.} for every weakly compactly generated Banach space Y , {\displaystyle Y,} every bounded linear operator from X
Grothendieck_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
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
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
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
Mathematics of real numbers and real functions
numbers and Euclidean spaces. Introductory real analysis is sometimes called advanced calculus, and studies limits, continuity, compactness, differentiation
Real_analysis
Concept in mathematics
"degenerate" examples of hyperbolic spaces are spaces with bounded diameter (for example finite or compact spaces) and the real line. Metric trees and
Hyperbolic_metric_space
Type of internal combustion engine cylinder head
having a depression in the top of each piston, namely: (i) it provides a compact space for combustion to begin, allowing an optimal flame front; and (ii) it
Heron_cylinder_head
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)
Connected non-abelian Lie group lacking nontrivial connected normal subgroups
symmetric spaces). The irreducible simply connected symmetric spaces are the real line, and exactly two symmetric spaces corresponding to each non-compact simple
Simple_Lie_group
Regular tiling of hyperbolic 3-space
order-5 cubic honeycomb is one of four compact regular space-filling tessellations (or honeycombs) in hyperbolic 3-space. With Schläfli symbol {4,3,5}, it
Order-5_cubic_honeycomb
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