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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 large cardinal in set theory
In mathematics, a weakly compact cardinal is a certain kind of cardinal number introduced by Erdős & Tarski (1961); weakly compact cardinals are large
Weakly_compact_cardinal
Type of topological space in mathematics
that compact subsets of Hausdorff spaces are closed, and closed subsets of compact spaces are compact. Spaces satisfying (1) are also called weakly locally
Locally_compact_space
Mathematical concept
vector space weakly closed (respectively, weakly compact, etc.) if they are closed (respectively, compact, etc.) with respect to the weak topology. Likewise
Weak_topology
Normed vector space that is complete
{\displaystyle X} has a weakly convergent subsequence. A weakly compact subset A {\displaystyle A} in ℓ 1 {\displaystyle \ell ^{1}} is norm-compact. Indeed, every
Banach_space
Type of convergence in Hilbert spaces
it converges weakly as well. Since every closed and bounded set is weakly relatively compact (its closure in the weak topology is compact), every bounded
Weak convergence (Hilbert space)
Weak_convergence_(Hilbert_space)
Theorem in mathematics
the theorem states that a weakly closed subset C {\displaystyle C} of a Banach space X {\displaystyle X} is weakly compact if and only if the dual norm
James's_theorem
Compactness theorem in Yang–Mills theory
Uhlenbeck's compactness theorem is a result about sequences of (weak Yang–Mills) connections with uniformly bounded curvature having weakly or uniformly
Uhlenbeck's compactness theorem
Uhlenbeck's_compactness_theorem
Type of topological space in mathematics
topological space X {\displaystyle X} is said to be limit point compact or weakly countably compact if every infinite subset of X {\displaystyle X} has a limit
Limit_point_compact
Concept in measure theory
{\displaystyle M} is sequential weak compactness. We say the family M {\displaystyle M} of probability measures is sequentially weakly compact if for every sequence
Tightness_of_measures
Type of topological space
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 and (weakly) locally
Σ-compact_space
model of ZFC worldly cardinals weakly and strongly inaccessible, α-inaccessible, and hyper inaccessible cardinals weakly and strongly Mahlo, α-Mahlo, and
List of large cardinal properties
List_of_large_cardinal_properties
Kind of large cardinal number
is a Mahlo cardinal. However, the first Woodin cardinal is not even weakly compact.p. 364 The hierarchy V α {\displaystyle V_{\alpha }} (known as the von
Woodin_cardinal
and B. J. Pettis, is a property of a Banach space stating that all weakly compact operators from this space into another Banach space are completely continuous
Dunford–Pettis_property
Index of articles associated with the same name
Compact cardinal may refer to: Weakly compact cardinal Subcompact cardinal Supercompact cardinal Strongly compact cardinal This set index article includes
Compact_cardinal
{\displaystyle K} a weakly compact subset of X {\displaystyle X} (that is, K {\displaystyle K} is compact when X {\displaystyle X} is endowed with the weak topology)
Krein–Smulian_theorem
Relates three different kinds of weak compactness in a Banach space
subsequence that is weakly convergent in X each sequence of elements of A has a weak cluster point in X the weak closure of A is weakly compact. A set A (in
Eberlein–Šmulian_theorem
Tree in set theory
κ is weakly compact then no κ-Aronszajn trees exist. Conversely, if κ is inaccessible and no κ-Aronszajn trees exist, then κ is weakly compact. An Aronszajn
Aronszajn_tree
Class of alternative set theories
strength of Morse–Kelley set theory with the proper class ordinal a weakly compact cardinal. The universal set is a proper set in this theory. The sets
Positive_set_theory
λ-compactness. A cardinal κ is weakly compact if and only if it is κ-compact; this was the original definition of that concept. Strong compactness implies
Strongly_compact_cardinal
Kind of large cardinal number
\setminus X\in {\mathcal {F}}} . This is similar to a characterization of weakly compact cardinals. More generally, κ {\displaystyle \kappa } is called n {\displaystyle
Ineffable_cardinal
Theorem in functional analysis
sets are weakly closed (Hahn–Banach theorem), norm-closures of convex bounded sets in Hilbert spaces or reflexive Banach spaces are weakly compact. Closed
Banach–Alaoglu_theorem
compact sets. X {\displaystyle X} is σ-compact and weakly locally compact. X {\displaystyle X} is Lindelöf and weakly locally compact. (where weakly locally
Exhaustion_by_compact_sets
Dual space topology of uniform convergence on some sub-collection of bounded subsets
{\displaystyle Y} carry their weak topologies. If G ′ {\displaystyle {\mathcal {G}}'} was the set of all convex balanced weakly compact equicontinuous subsets
Polar_topology
mathematics, a weak Hausdorff space or weakly Hausdorff space is a topological space where the image of every continuous map from a compact Hausdorff space
Weak_Hausdorff_space
weakly compact, that is, the image of a bounded subset of X {\displaystyle X} is a weakly compact subset of Y . {\displaystyle Y.} for every weakly compactly
Grothendieck_space
Locally convex topological vector space
are weakly closed, it follows from the third property that closed bounded convex subsets of a reflexive space X {\displaystyle X} are weakly compact. Thus
Reflexive_space
Category used in algebraic topology
In mathematics, the category of compactly generated weak Hausdorff spaces, CGWH, is a category used in algebraic topology as an alternative to the category
Category of compactly generated weak Hausdorff spaces
Category_of_compactly_generated_weak_Hausdorff_spaces
Axiomatic set theory devised by W.V.O. Quine
counterexamples. w e a k l y c o m p a c t {\displaystyle {\mathsf {weakly\ compact}}} see weakly compact cardinal. m e a s u r a b l e {\displaystyle {\mathsf {measurable}}}
New_Foundations
Magnetic audio tape recording format
The cassette tape, officially named the Compact Cassette, and also known as audio cassette, or simply tape or cassette, is an analog magnetic tape recording
Cassette_tape
Mathematical technique used in proof theory
(\varepsilon _{K+1})} , where K {\displaystyle K} refers to the first weakly compact, due to (Rathjen 1993) K P + Π ω − R e f {\displaystyle {\mathsf {KP}}+\Pi
Ordinal_analysis
Smallest convex set containing a given set
closed convex hull of a weakly compact subset of a Banach space (a subset that is compact under the weak topology) is weakly compact. An extreme point of
Convex_hull
Type of mathematical space
also equivalent to compactness for first-countable uniform spaces). (X, d) is limit point compact (also called weakly countably compact); that is, every
Compact_space
Ordinals in mathematics and set theory
{\displaystyle \varepsilon _{K+1}} , where K {\displaystyle K} is the first weakly compact (= Π 1 1 {\displaystyle \Pi _{1}^{1}} -indescribable) cardinal. This
Large_countable_ordinal
Partial order with well-ordered predecessors
(the converse does not hold). One of the equivalent ways to define a weakly compact cardinal is that it is an inaccessible cardinal κ {\displaystyle \kappa
Tree_(set_theory)
pointwise compact and contained in c0, it is weakly compact in c0. Let V be the closed convex hull of K in c0. It is also a weakly compact set in c0.
Tsirelson_space
Type of vector space in math
any orthonormal sequence {fn} converges weakly to 0, as a consequence of Bessel's inequality. Every weakly convergent sequence {xn} is bounded, by the
Hilbert_space
Large cardinal number
is false. The least ω {\displaystyle \omega } -Erdős cardinal is not weakly compact,p. 39. nor is the least ω 1 {\displaystyle \omega _{1}} -Erdős cardinal
Erdős_cardinal
topology, the topology of uniform convergence on all absolutely convex weakly compact subsets of X ′ {\displaystyle X'} . Given a dual pair ( X , X ′ ) {\displaystyle
Dual_topology
Type of continuous linear operator
In functional analysis, a branch of mathematics, a compact operator is a linear operator that behaves, in several important respects, like a finite-dimensional
Compact_operator
Property of topological spaces
space of a weakly locally compact space is CG-1. Conversely, every CG-1 space X {\displaystyle X} is the quotient space of a weakly locally compact space,
Compactly_generated_space
Feature of certain mathematical spaces
compact embedding theorems. When an embedding is not compact, it may possess a related, but weaker, property of cocompactness. Let X {\displaystyle X}
Compact_embedding
Order type of the set of all recursive ordinals
<\gamma } . An ordinal α {\displaystyle \alpha } is called recursively weakly compact if it is Π 3 {\displaystyle \Pi _{3}} -reflecting, or equivalently,
Nonrecursive_ordinal
Romanian-American mathematician (1935–2025)
modification). Furthermore, by applying a lifting to a ‘weakly’ measurable function with values in a weakly compact set of a Banach space, one obtains a strongly
Alexandra_Bellow
Mathematical function
y'\right),} where S and T are respectively some weakly closed and equicontinuous (hence weakly compact) subsets of the duals X ′ {\displaystyle X^{\prime
Integral_linear_operator
Classification in astronomy
In astronomy, the term compact object (or compact star) refers collectively to white dwarfs, neutron stars, and black holes. It could also include exotic
Compact_object
Vopěnka's principle holds for Vκ weakly 1. A weakly inaccessible cardinal is a regular weak limit cardinal 2. A weakly compact cardinal is a cardinal κ (usually
Glossary_of_set_theory
Vector space in mathematics
result: Theorem. A bounded linear operator between two Banach spaces is weakly compact if and only if it factors through a reflexive space. The space ℓ q used
Interpolation_space
Vector space of infinite sequences
converges weakly in this space. If K {\displaystyle K} is a subset of this space, then the following are equivalent: K {\displaystyle K} is compact;
Sequence_space
Fluorescent lamps with folded tubes, often with built-in ballast
Compact fluorescent lamp (CFL) examples A compact fluorescent lamp (CFL), also called compact fluorescent light, energy-saving light and compact fluorescent
Compact_fluorescent_lamp
Maximal proper filter
topological vector space (TVS) is relatively compact in the weak-* topology (that is, it is contained in some weak-* compact set). The polar of any neighborhood
Ultrafilter_on_a_set
Functional analysis concept
{\displaystyle H} , the closed unit ball B {\displaystyle B} is weakly compact. Also, the compactness of T {\displaystyle T} means (see above) that T {\displaystyle
Compact operator on Hilbert space
Compact_operator_on_Hilbert_space
Inaccessible cardinal Mahlo cardinal Measurable cardinal Supercompact cardinal Weakly compact cardinal Linear partial information Multiset Musical set theory Ordinal
List_of_set_theory_topics
Japanese multinational imaging corporation
July 2020. Retrieved 29 July 2020. "Canon Q1 operating profit dips on weaker compact camera sales". Reuters. 24 April 2013. Archived from the original on
Canon_Inc.
(pseudo-)Riemannian manifold whose geodesics are reversible
definition of symmetric space to that of weakly symmetric Riemannian space, or in current terminology weakly symmetric space. These are defined as Riemannian
Symmetric_space
norm-closed. These include but are not limited to the following. Compact operators Weakly compact operators Finitely strictly singular operators Strictly singular
Operator_ideal
-topology has the Heine–Borel property (i.e. weakly closed and bounded subsets of X {\displaystyle X} are weakly compact). Every semi-Montel space is semi-reflexive
Semi-reflexive_space
Hypothetical particles that may constitute dark matter
Weakly interacting massive particles (WIMPs) are hypothetical particles that are one of the proposed candidates for dark matter. These particles would
Weakly interacting massive particle
Weakly_interacting_massive_particle
Frederick 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
Theorem in measure theory
Prokhorov's theorem relates tightness of measures to relative compactness (and hence weak convergence) in the space of probability measures. It is credited
Prokhorov's_theorem
Theorem in functional analysis
_{k}} But A is compact, therefore the function f(x) = (Ax, x) is weakly continuous. Furthermore, any bounded set in H is weakly compact. This lets us replace
Min-max_theorem
Mathematical concept
converge weakly (or in distribution or in law) to the random variable X: Ω → X as n → ∞ if the sequence of pushforward measures (Xn)∗(P) converges weakly to
Convergence_of_measures
Being equally consistent
\omega _{2}} -Aronszajn trees is equiconsistent with the existence of a weakly compact cardinal. Large cardinal property *Kunen, Kenneth (2011), Set theory
Equiconsistency
consistent with V=L including: the least Mahlo cardinal, the least weakly compact cardinal, the least unfoldable cardinal, the least ineffable cardinal
Silver_cardinal
Mathematics of convex functions and sets
reflexive Banach space, closed bounded convex sets have useful weak compactness properties, making weak lower semicontinuity a standard tool in variational problems
Convex_analysis
is weakly harmonic if and only if it is harmonic. Thus weakly harmonic is actually equivalent to the seemingly stronger harmonic condition. Weak solution
Weakly_harmonic_function
Relation among continuous functions
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 bounded and equicontinuous
Equicontinuity
unless X is finite-dimensional. Each order interval in an AL-space is weakly compact. The strong dual of an AL-space is an AM-space with unit. The continuous
Abstract_L-space
Extension of ideas in combinatorics to infinite sets
cardinal properties can be defined using this notation. In particular: Weakly compact cardinals κ {\displaystyle \kappa } are those that satisfy κ → ( κ )
Infinitary_combinatorics
Pre-pressed compact disc containing computer data
CD-ROM (/ˌsiːdiːˈrɒm/, compact disc read-only memory) is a type of read-only memory consisting of a pre-pressed optical compact disc that contains data
CD-ROM
Reflecting cardinals were introduced by (Mekler & Shelah 1989). Every weakly compact cardinal is a reflecting cardinal, and is also a limit of reflecting
Reflecting_cardinal
\varphi \circ f} is a measurable map. A weakly measurable map f : X → V {\displaystyle f:X\to V} is said to be weakly integrable on X {\displaystyle X} if
Pettis_integral
Data format used for audio compact discs
Compact Disc Digital Audio (CDDA or CD-DA), also known as Digital Audio Compact Disc or simply as Audio CD, is the standard format for audio compact discs
Compact_Disc_Digital_Audio
Generalization of compactness
related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally
Totally_bounded_space
Superstrong cardinal Totally indescribable cardinal Weakly compact cardinal Weakly hyper-Woodin cardinal Weakly inaccessible cardinal Woodin cardinal Unfoldable
List of mathematical logic topics
List_of_mathematical_logic_topics
it weakly complete. The Arzelà–Ascoli theorem holds: A subset K {\displaystyle K} of C ( X ) {\displaystyle {\mathcal {C}}(X)} is relatively compact if
Space of continuous functions on a compact space
Space_of_continuous_functions_on_a_compact_space
Set-theoretic function
α-weakly inaccessible if it is uncountable, regular and it is a limit of γ-weakly inaccessible cardinals for γ < α. Let I(α, 0) be the first α-weakly inaccessible
Ordinal_collapsing_function
Logic that allows infinitely long proofs
strongly complete. A cardinal κ ≠ ω {\displaystyle \kappa \neq \omega } is weakly compact when for every theory T in L κ , κ {\displaystyle L_{\kappa ,\kappa
Infinitary_logic
Function that "converges" to periodicity
a compact group and a finite-dimensional vector space. A function on a locally compact group is called weakly almost periodic if its orbit is weakly relatively
Almost_periodic_function
Generalisation of the derivative of a function
all infinitely differentiable functions φ {\displaystyle \varphi } with compact support in U {\displaystyle U} . Here D α φ {\displaystyle D^{\alpha }\varphi
Weak_derivative
Topology where a set is open if it contains a particular point
point thus if X if infinite it is not weakly countably compact. Locally compact but not locally relatively compact. If x ∈ X {\displaystyle x\in X} , then
Particular_point_topology
Auxiliary functions used to probe equations, distributions, and weak formulations
function φ {\displaystyle \varphi } : U → R is said to have compact support if there exists a compact subset K of U such that φ {\displaystyle \varphi } (x)
Test_function
On when a space equals the closed convex hull of its extreme points
{\displaystyle 0<p<1.} Linearity is also needed, because the statement fails for weakly compact convex sets in CAT(0) spaces, as proved by Nicolas Monod (2016). However
Krein–Milman_theorem
Polish mathematician (1948–2025)
J.; Figiel, T.; Johnson, W. B.; Pelczynski, A. (1974). "Factoring weakly compact operators" (PDF). Journal of Functional Analysis. 17 (3): 311–327. doi:10
Tadeusz_Figiel
k is κ-strongly unfoldable, and κ-unfoldable, if and only if it is weakly compact. A κ+ω-unfoldable cardinal is indescribable and preceded by a stationary
Unfoldable_cardinal
decidable; decidability for ω2 is independent of ZFC (assuming Con(ZFC + weakly compact cardinal)). Also, an ordinal is definable using monadic second-order
S2S_(mathematics)
Concept in order theory
unit ball is a non-empty and weakly compact (i.e. σ ( X ′ , X ) {\displaystyle \sigma \left(X^{\prime },X\right)} -compact) subset of X ′ {\displaystyle
Abstract_m-space
always induces the weak* topology on the space of Radon probability measures over X {\displaystyle X} --- which is weak* compact. Rieffel worked out
Spectral_triple
Faint neutron star at the center of a supernova remnant
A central compact object (CCO) is an x-ray source found near the center of a young, nearby supernova remnant (SNR). Given the x-ray flux and spectra of
Central_compact_object
Locally compact topological group with an invariant averaging operation
representations are weakly contained in the left regular representation λ on L2(G). Trivial representation. The trivial representation of G is weakly contained
Amenable_group
countable certifiability, Kc would correctly compute the successors of all weakly compact and singular strong limit cardinals correctly. If V is closed under
Core_model
Solution concept of a non-cooperative game
unlimited private communication. In fact, strong Nash equilibrium has to be weakly Pareto efficient. As a result of these requirements, strong Nash is too
Nash_equilibrium
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
Dual pair of vector spaces
well-defined. If F {\displaystyle F} is weakly continuous then t F : Z → Y {\displaystyle {}^{t}F:Z\to Y} is weakly continuous, meaning that t F : ( Z ,
Dual_system
space Eberlein–Šmulian theorem – Relates three different kinds of weak compactness in a Banach space James' theorem – Theorem in mathematicsPages displaying
Bishop–Phelps_theorem
Mathematical Banach space
its dual space X∗ has weak-∗-slices of arbitrarily small diameter. X is Asplund if and only if every non-empty weakly-∗ compact convex subset of the dual
Asplund_space
and |x| = |y|. K computes the successors of singular and weakly compact cardinals correctly (Weak Covering Property). Moreover, if |κ| > ω1, then cofinality((κ+)K) ≥ |κ|
Covering_lemma
Semi-automatic pistol made by ČZUB
CZ 85 Compact A limited production compact CZ 85 with under-barrel accessory rail and chambered in .40 S&W. Identical to the current CZ 75 compact in .40
CZ_75
{\displaystyle H} is a weakly closed equicontinuous disk in X ′ {\displaystyle X^{\prime }} (this implies that H {\displaystyle H} is weakly compact) and let U :=
Auxiliary_normed_space
Blue gemstone from Dominican Republic
Perfect on {100} and {001} Fracture Uneven Tenacity Brittle; tough when compact Mohs scale hardness 4.5 - 5 Luster Silky, subvitreous Diaphaneity Translucent
Larimar
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