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Topological space which is a generalization of certain compact spaces
(1944). Every compact space is paracompact. Every paracompact Hausdorff space is normal, and a Hausdorff space is paracompact if and only if it admits
Paracompact_space
Theorem in topology
regular topological space (in fact, for a T1-space) to be paracompact. A family E i {\displaystyle E_{i}} of subsets of a topological space is said to be closure-preserving
Michael's theorem on paracompact spaces
Michael's_theorem_on_paracompact_spaces
Topological space associated to a vector bundle
topology is a topological space associated to a vector bundle, over any paracompact space. One way to construct this space is as follows. Let p : E →
Thom_space
space is said to be a-paracompact if every open cover of the space has a locally finite refinement. In contrast to the definition of paracompactness,
A-paracompact_space
Type of topological space
the above examples, all paracompact Hausdorff spaces are normal, and all paracompact regular spaces are normal; All paracompact topological manifolds are
Normal_space
Type of mathematical space
compact sets Lindelöf space Metacompact space Noetherian topological space Orthocompact space Paracompact space Precompact set - also called totally bounded
Compact_space
topology. Every Menger space is a D-space. A subspace of a topological linearly ordered space is a D-space iff it is a paracompact space. van Douwen, E.; Pfeffer
D-space
Mathematical set with some added structure
Minkowski space Müntz space Normed space Paracompact space Perfectoid space Planar space Polish space Probability space Projective space Proximity space Quadratic
Space_(mathematics)
Topological space that is homeomorphic to a metric space
space to be metrizable. Metrizable spaces inherit all topological properties from metric spaces. For example, they are Hausdorff paracompact spaces (and
Metrizable_space
Topological space with a point-finite open refinement for every cover
topological spaces: Every paracompact space is metacompact. This implies that every compact space is metacompact, and every metric space is metacompact
Metacompact_space
Type of topological space
regular Lindelöf space is normal. Every regular Lindelöf space is paracompact. A countable union of Lindelöf subspaces of a topological space is Lindelöf.
Lindelöf_space
Tessellation of convex uniform polyhedron cells
hyperbolic space are tessellations of convex uniform polyhedron cells. In 3-dimensional hyperbolic space there are 23 Coxeter group families of paracompact uniform
Paracompact uniform honeycombs
Paracompact_uniform_honeycombs
pseudocompact (see Engelking, p. 153). Compact space Paracompact space Normal space Pseudocompact space Tychonoff space Gillman, Leonard; Jerison, Meyer, "Rings
Realcompact_space
finite. Every countably compact paracompact space is compact. More generally, every countably compact metacompact space is compact. Every countably compact
Countably_compact_space
Construction for vector bundles
every vector bundle over paracompact spaces a line bundle. Its name comes from using the determinant on their classifying spaces. Determinant line bundles
Determinant_line_bundle
Mathematical method
sufficient for the existence of a continuous selection: X is a paracompact space; Y is a Banach space; F is lower hemicontinuous; for all x in X, the set F(x)
Selection_theorem
Topological concept for collections of sets
topological space is also point-finite. A topological space in which every open cover admits a locally finite open refinement is called a paracompact space. Every
Point-finite_collection
uniformity on X is complete. Every regular paracompact space (in particular, every Hausdorff paracompact space) is completely uniformizable. (Shirota's
Completely uniformizable space
Completely_uniformizable_space
Topological space with a bounded image under any continuous function to R
483-496, 1966. [2] Compact space Paracompact space Normal space Realcompact space Metacompact space Orthocompact space Tychonoff space Engelking, Ryszard (1968)
Pseudocompact_space
French mathematician (1906–1992)
theorem Dieudonné complete space Dieudonné determinant Dieudonné plank Dieudonné module Dieudonné's theorem Paracompact space Awards Lester R. Ford Award
Jean_Dieudonné
Generalization of compactness
{\displaystyle K} is complete. Compact space Locally compact space Measure of non-compactness Orthocompact space Paracompact space Relatively compact subspace Sutherland
Totally_bounded_space
Theorem in functional analysis
following: Michael Selection Theorem—Let X be a paracompact space and Y be a separable Banach space. Let F : X → Y {\displaystyle F\colon X\to Y} be
Michael_selection_theorem
Exact homotopy case
classifying space for the unitary group U(n) is a space BU(n) together with a universal bundle EU(n) such that any hermitian bundle on a paracompact space X is
Classifying_space_for_U(n)
Topological concept
property of collections of subsets of a topological space. It is fundamental in the study of paracompactness and topological dimension. Note that the term locally
Locally_finite_collection
Property of topological spaces stronger than normality
collectionwise normal space is collectionwise Hausdorff. A collectionwise normal space is normal. A Hausdorff paracompact space is collectionwise normal
Collectionwise_normal_space
E_{x}\otimes _{\mathbb {R} }\mathbb {C} } . Any complex vector bundle over a paracompact space admits a hermitian metric. The basic invariant of a complex vector
Complex_vector_bundle
Topological space with an interior-preserving refinement for every open cover
Hence, we have the following: every metacompact space, and in particular, every paracompact space, is orthocompact. Useful theorems: Orthocompactness
Orthocompact_space
Type of topological space
to nonregular Hausdorff spaces. There are many situations where another condition of topological spaces (such as paracompactness or local compactness) will
Hausdorff_space
Mathematical space with a notion of distance
Introduction to Metric Spaces and Fixed Point Theory, page 14, John Wiley & Sons Rudin, Mary Ellen. A new proof that metric spaces are paracompact Archived 2016-04-12
Metric_space
facts are true about mesocompactness: Every compact space, and more generally every paracompact space is mesocompact. This follows from the fact that any
Mesocompact_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
Inputs for which a function's value is non-zero
{\displaystyle Y} in Φ {\displaystyle \Phi } is, with the subspace topology, a paracompact space; and has some Z {\displaystyle Z} in Φ {\displaystyle \Phi } which
Support_(mathematics)
Type of topological space
metrizable nor paracompact. Since metrizability is such a desirable property for a topological space, it is common to add paracompactness to the definition
Topological_manifold
Mathematics concept
Every compact space is feebly compact. Every feebly compact paracompact space is compact.[citation needed] Every feebly compact space is pseudocompact
Feebly_compact_space
hyperbolic 5-space, the 5-orthoplex honeycomb is one of five paracompact regular space-filling tessellations (or honeycombs). It is paracompact because it
5-orthoplex_honeycomb
Topological space characterized by sequences
"Topological vector space". Encyclopedia of Mathematics. Retrieved September 6, 2020. It is a Montel space, hence paracompact, and so normal. Trèves
Sequential_space
Property of topological space
Hausdorff spaces. There are many situations where another condition of topological spaces (such as normality, pseudonormality, paracompactness, or local
Regular_space
Topologically invariant definition of the dimension of a space
dimension of a normal space is less than or equal to the large inductive dimension. The covering dimension of a paracompact Hausdorff space X {\displaystyle
Lebesgue_covering_dimension
mathematical field of general topology, a Dowker space is a topological space that is T4 but not countably paracompact. They are named after Clifford Hugh Dowker
Dowker_space
Mathematical technique for vector bundles
vector bundle of rank n {\displaystyle n} over a paracompact space X {\displaystyle X} . There exists a space Y = F l ( E ) {\displaystyle Y=Fl(E)} , called
Splitting_principle
Topological space in mathematics
.} This space is not compact, but the union of any countable set of compact subspaces has compact closure. Some examples of non-paracompact manifolds
Long_line_(topology)
Normal space – Type of topological space – a topological space in which every two disjoint closed sets have disjoint open neighborhoods Paracompact space –
Paranormal_space
Quotient of a weakly contractible space by a free action
action of G. It has the property that any G principal bundle over a paracompact manifold is isomorphic to a pullback of the principal bundle E G → B
Classifying_space
Concept in topology
topology is used instead. If X {\displaystyle X} is a paracompact and Y {\displaystyle Y} is a metric space, then the Whitney topology has a basic open set
Mapping_space
Mathematical approach
behaviour of paracompactness, with arbitrary products of paracompact locales being paracompact, which is not true for paracompact spaces, or the fact
Pointless_topology
Soviet mathematician (1896–1982)
Alexandrov. In fact, this proved the paracompact nature of separable metric spaces (although the term "paracompact space" was introduced by Jean Dieudonné
Pavel_Alexandrov
every normal metacompact space is a shrinking space. In particular, every Hausdorff paracompact space is a shrinking space. These facts are particularly
Shrinking_space
Branch of mathematics
lifting. If E {\displaystyle E} is a principal G-bundle over a paracompact space, that is, a space with a free and transitive (topological) group action of
Homotopy_theory
Topics referred to by the same term
Hausdorff spaces the property of being a paracompact space and being a fully normal space are equivalent, or its immediate corollary that metric spaces are
Stone's_theorem
geometry of hyperbolic 3-space, the square tiling honeycomb is one of 11 paracompact regular honeycombs. It is called paracompact because it has infinite
Square_tiling_honeycomb
Topological space whose topology has a countable base
second-countable, Hausdorff regular space is metrizable. It follows that every such space is completely normal as well as paracompact. Second-countability is therefore
Second-countable_space
American mathematician
also known in topology for the Michael line, a paracompact space whose product with the topological space of the irrational numbers is not normal. He wrote
Ernest_Michael
hyperbolic 3-space, the order-6 tetrahedral honeycomb is a paracompact regular space-filling tessellation (or honeycomb). It is paracompact because it has
Order-6_tetrahedral_honeycomb
Cohomology with real coefficients computed using differential forms
^{k})} vanish for i > 0 {\textstyle i>0} since all fine sheaves on paracompact spaces are acyclic. So the long exact cohomology sequences themselves ultimately
De_Rham_cohomology
Smooth manifold with an inner product on each tangent space
locally Euclidean topological space, for this result it is necessary to use that smooth manifolds are Hausdorff and paracompact. The reason is that the proof
Riemannian_manifold
Vector bundle existing over a Grassmannian
_{n}^{\mathbb {R} }(X)} for any paracompact space X. Since G n {\displaystyle G_{n}} is the direct limit of compact spaces, it is paracompact and so there is a unique
Tautological_bundle
3-space can be called hyperbolic honeycombs. There are 15 hyperbolic honeycombs in H3, 4 compact and 11 paracompact. There are also 11 paracompact H3
List_of_regular_polytopes
Topics referred to by the same term
Littlewood subordination theorem Subordinate partition of unity in paracompact space Subordinator (disambiguation) This disambiguation page lists articles
Subordination
Characteristic classes of vector bundles
{\displaystyle \pi \colon E\to B} be a complex vector bundle over a paracompact space B. Thinking of B as being embedded in E as the zero section, let B
Chern_class
Topological vector spaces
three spaces, are complete nuclear Montel bornological spaces, which implies that all six of these locally convex spaces are also paracompact reflexive
Spaces of test functions and distributions
Spaces_of_test_functions_and_distributions
Tiling of euclidean or hyperbolic space of three or more dimensions
honeycombs and polychora. The 4 compact and 11 paracompact regular hyperbolic honeycombs and many compact and paracompact uniform hyperbolic honeycombs have been
Honeycomb_(geometry)
Regular paracompact honeycomb
tiling honeycomb is one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is paracompact because it has cells composed of an infinite
Hexagonal_tiling_honeycomb
Topics referred to by the same term
after Ernest Michael, can mean: Michael's theorem on paracompact spaces saying that a regular space in which each open cover has a refinement by a countable
Michael's_theorem
5-space, the 16-cell honeycomb honeycomb is one of five paracompact regular space-filling tessellations (or honeycombs). It is called paracompact because
16-cell_honeycomb_honeycomb
geometry of hyperbolic 3-space, the order-4 square tiling honeycomb is one of 11 paracompact regular honeycombs. It is paracompact because it has infinite
Order-4 square tiling honeycomb
Order-4_square_tiling_honeycomb
countably paracompact space Y, is X metrizable and sigma-locally compact? The answers were believed to be affirmative. Here a normal P-space Y is characterised
Morita_conjectures
Certain topology in mathematics
and countably compact, but not compact or paracompact Any ordinal number can be viewed as a topological space by endowing it with the order topology (indeed
Order_topology
5-space, the 24-cell honeycomb honeycomb is one of five paracompact regular space-filling tessellations (or honeycombs). It is called paracompact because
24-cell_honeycomb_honeycomb
Frequently-cited counterexample in topology
ISBN 0-387-90125-6. Robert Sorgenfrey, "On the topological product of paracompact spaces", Bull. Amer. Math. Soc. 53 (1947) 631–632. Steen, Lynn Arthur; Seebach
Sorgenfrey_plane
Infinite regular skew polyhedron
apeirohedra in hyperbolic 3-space with Petrie and Coxeters definition, discovering 31 regular skew apeirohedra with compact or paracompact symmetry. In 1977 Grünbaum
Regular_skew_apeirohedron
List of concrete topologies and topological spaces
locally regular space but not a semiregular space. Prüfer manifold − A Hausdorff 2-dimensional real analytic manifold that is not paracompact. Real projective
List_of_topologies
The order-4 octahedral honeycomb is a regular paracompact honeycomb in hyperbolic 3-space. It is paracompact because it has infinite vertex figures, with
Order-4_octahedral_honeycomb
Canadian topologist (1912–1982)
introduced the concept of countably paracompact spaces. In the same article, Dowker conjectured that so-called Dowker spaces could not exist. This conjecture
Clifford_Hugh_Dowker
honeycomb arises as one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is paracompact because it has cells composed of an infinite
Order-4 hexagonal tiling honeycomb
Order-4_hexagonal_tiling_honeycomb
Regular geometrical object in hyperbolic space
order-6 dodecahedral honeycomb is one of 11 paracompact regular honeycombs in hyperbolic 3-space. It is paracompact because it has vertex figures composed
Order-6 dodecahedral honeycomb
Order-6_dodecahedral_honeycomb
Concept in topology
is not paracompact (hence not metrizable), like the long line. One should note however that there are plenty of spaces that are Baire spaces without
Baire_space
Set of topological invariants
that if X is a paracompact space, this map is a bijection. This is the reason why we call infinite Grassmannians the classifying spaces of vector bundles
Stiefel–Whitney_class
In the geometry of hyperbolic 5-space, the order-4 24-cell honeycomb honeycomb is one of five paracompact regular space-filling tessellations (or honeycombs)
Order-4 24-cell honeycomb honeycomb
Order-4_24-cell_honeycomb_honeycomb
Category theory
subcategory of free modules. The category of vector bundles over any paracompact space is the Karoubi envelope of its full subcategory of trivial bundles
Karoubi_envelope
Mathematics glossary
telescope Thom 1. René Thom. 2. If E is a vector bundle on a paracompact space X, then the Thom space Th ( E ) {\displaystyle {\text{Th}}(E)} of E is obtained
Glossary of algebraic topology
Glossary_of_algebraic_topology
honeycomb is one of 11 paracompact regular space-filling tessellations (or honeycombs) in hyperbolic 3-space. It is called paracompact because it has infinite
Triangular_tiling_honeycomb
Tiling of hyperbolic 3-space by uniform polyhedra
diagrams for each family. Honeycombs are divided between compact and paracompact forms defined by Coxeter groups, the first category only including finite
Uniform honeycombs in hyperbolic space
Uniform_honeycombs_in_hyperbolic_space
Construct in mathematics
the algebra of compactly supported complex valued functions on a paracompact space X {\displaystyle X} pg 3. Given a cover U = { U i } {\displaystyle
Gerbe
Example of a metacompact topological space that is not paracompact
a specific topological space introduced by Dieudonné (1944). It is an example of a metacompact space that is not paracompact. The notion has since been
Dieudonné_plank
honeycomb arises as one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is paracompact because it has cells composed of an infinite
Order-5 hexagonal tiling honeycomb
Order-5_hexagonal_tiling_honeycomb
Type of topological space
"Topological vector space". Encyclopedia of Mathematics. Retrieved September 6, 2020. It is a Montel space, hence paracompact, and so normal. Gabriyelyan
Fréchet–Urysohn_space
Branch of topology
{\displaystyle M} for a metric space if it is clear from the context what metric is used. Every metric space is paracompact and Hausdorff, and thus normal
General_topology
Paracompact space Locally compact space Compactly generated space Axiom of countability Sequential space First-countable space Second-countable space
List of general topology topics
List_of_general_topology_topics
hyperbolic tilings. Hyperbolic tilings can also be divided between compact, paracompact and divergent cases. The uniform tilings are the simplest application
Uniform tiling symmetry mutations
Uniform_tiling_symmetry_mutations
hyperbolic 4-space, the cubic honeycomb honeycomb is one of two paracompact regular space-filling tessellations (or honeycombs). It is called paracompact because
Cubic_honeycomb_honeycomb
tiling honeycomb is one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is paracompact because it has cells with an infinite
Order-6 hexagonal tiling honeycomb
Order-6_hexagonal_tiling_honeycomb
In the geometry of hyperbolic 3-space, the tetrahedral-square tiling honeycomb is a paracompact uniform honeycomb, constructed from tetrahedron, cuboctahedron
Tetrahedral-square tiling honeycomb
Tetrahedral-square_tiling_honeycomb
Mathematical property of a space
countable subcover. Paracompact. A space is paracompact if every open cover has an open locally finite refinement. Paracompact Hausdorff spaces are normal. Locally
Topological_property
Function whose values are sets (mathematics)
Michael selection theorem, which provides another characterisation of paracompact spaces. Other selection theorems, like Bressan-Colombo directional continuous
Set-valued_function
refinement. Paracompact A space is paracompact if every open cover has a locally finite open refinement. Paracompact implies metacompact. Paracompact Hausdorff
Glossary_of_general_topology
{\displaystyle X} , an exhaustion by compact sets can be used to show the space is paracompact. Indeed, suppose we have an increasing sequence V 1 ⊂ V 2 ⊂ ⋯ {\displaystyle
Exhaustion_by_compact_sets
Mathematical object in sheaf cohomology
over a smooth (paracompact Hausdorff) manifold, or modules over these sheaves of rings. Also, fine sheaves over paracompact Hausdorff spaces are soft and
Injective_sheaf
hyperbolic 4-space, the order-4 24-cell honeycomb is one of two paracompact regular space-filling tessellations (or honeycombs). It is called paracompact because
Order-4_24-cell_honeycomb
Rule in mathematics
{\displaystyle X\times Y} has property P. Every separable productively paracompact space is U fin ( O , Γ ) {\displaystyle {\text{U}}_{\text{fin}}(\mathbf
Selection_principle
Geometrical concept
5-space, the tesseractic honeycomb honeycomb is one of five paracompact regular space-filling tessellations (or honeycombs). It is called paracompact because
Tesseractic honeycomb honeycomb
Tesseractic_honeycomb_honeycomb
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