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Embedding a topological space into a compact space as a dense subset
In mathematics, in general topology, compactification is the process or result of making a topological space into a compact space. A compact space is a
Compactification (mathematics)
Compactification_(mathematics)
Concept in topology
adding points so that certain kinds of limits exist. The Stone–Čech compactification of a space provides the most extensive such enlargement: it adds enough
Stone–Čech_compactification
Topics referred to by the same term
Look up compactification in Wiktionary, the free dictionary. Compactification may refer to: Compactification (mathematics), making a topological space
Compactification
A compactification of T1 topological spaces
In mathematics, the Wallman compactification, generally called Wallman–Shanin compactification is a compactification of T1 topological spaces that was
Wallman_compactification
Way to extend a non-compact topological space
is called the one-point compactification or Alexandroff compactification. The advantages of the Alexandroff compactification lie in its simple, often
Alexandroff_extension
In mathematics, the Bohr compactification of a topological group G is a compact Hausdorff topological group H that may be canonically associated to G.
Bohr_compactification
Configuration space
Fulton–MacPherson compactification by A. Voronov. Fulton, W.; MacPherson, R. (1994). "Compactification of configuration spaces". Annals of Mathematics. 139: 183–225
Fulton–MacPherson compactification
Fulton–MacPherson_compactification
2D surface which extends indefinitely
hypersurface in three-dimensional Minkowski space.) The one-point compactification of the plane is homeomorphic to a sphere (see stereographic projection);
Plane_(mathematics)
wonderful compactification of a variety acted on by an algebraic group G {\displaystyle G} is a G {\displaystyle G} -equivariant compactification such that
Wonderful_compactification
Concept of mathematics in convex analysis
In mathematics, specifically in convex analysis, the convex compactification is a compactification which is simultaneously a convex subset in a locally
Convex_compactification
In mathematics, the Baily–Borel compactification is a compactification of a quotient of a Hermitian symmetric space by an arithmetic group, introduced
Baily–Borel_compactification
Mathematical concept
the topological space of the real numbers, producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us the
Infinity
"Main problem and main results". Toroidal Compactification of Siegel Spaces. Lecture Notes in Mathematics. Vol. 812. Springer. pp. 7–11. doi:10.1007/BFb0091053
List of letters used in mathematics, science, and engineering
List_of_letters_used_in_mathematics,_science,_and_engineering
Adding a point at each end yields a compactification of the original space, known as the end compactification. The notion of an end of a topological
End_(topology)
Mathematical measure space associated to a random walk
compact image for the topology of pointwise convergence, and the Martin compactification is the closure of this image. A point γ ∈ Γ {\displaystyle \gamma \in
Poisson_boundary
Theory of subatomic structure
observed in experiments. Compactification is one way of modifying the number of dimensions in a physical theory. In compactification, some of the extra dimensions
String_theory
In mathematics, the prime end compactification is a method to compactify a topological disc (i.e. a simply connected open set in the plane) by adding
Prime_end
Mathematical concept
embedding Tevelev, Jenia (2007-08-07). "Compactifications of subvarieties of tori". American Journal of Mathematics. 129 (4): 1087–1104. arXiv:math/0412329
Tropical_compactification
Parametrizes complex structures on a surface
continuous action on this compactification. Gardiner & Masur (1991) considered a compactification similar to the Thurston compactification, but using extremal
Teichmüller_space
Mapping equal to its square under mapping composition
projected point for P. The correspondence makes the sphere a one-point compactification for the plane when a point at infinity is included to correspond to
Projection_(mathematics)
Type of mathematical space
Terence (2008). "Compactness and compactification". In Gowers, Timothy (ed.). The Princeton Companion to Mathematics. Princeton University Press. pp. 169–170
Compact_space
Mathematical symbol representing infinity
and the point added to a topological space to form its one-point compactification. In measure theory, the value of a measure is often taken as an extended
Infinity_symbol
Generalization of mass, length, area and volume
limits, the dual of L ∞ {\displaystyle L^{\infty }} and the Stone–Čech compactification. All these are linked in one way or another to the axiom of choice
Measure_(mathematics)
Topics referred to by the same term
Czech), nation and ethnic group Stone–Čech compactification, mathematical technique Čech cohomology, mathematical theory All pages with titles containing
Čech_(disambiguation)
Theory of strings with supersymmetry
occurring as a result of a Kaluza–Klein compactification of 11D M-theory that contains membranes. Because compactification of a geometric theory produces extra
Superstring_theory
248-dimensional exceptional simple Lie group
In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same
E8_(mathematics)
Framework of superstring theory
observed in experiments. Compactification is one way of modifying the number of dimensions in a physical theory. In compactification, some of the extra dimensions
M-theory
Space of complex matrices with positive definite imaginary part
Jr.; Borel, Armand (1966). "Compactification of arithmetic quotients of bounded symmetric domains". Annals of Mathematics. Second Series. 84 (3): 442–528
Siegel_upper_half-space
{\overline {\mathcal {T}}}} is compact: it is called the Thurston compactification of the Teichmüller space. The boundary T ¯ ∖ T {\displaystyle {\overline
Thurston_boundary
American mathematician (1930–2013)
Gale 2004 Baily-Borel Compactification, Encyclopedia of Mathematics "Putnam Competition Individual and Team Winners". Mathematical Association of America
Walter_Lewis_Baily_Jr.
Mathematical object studied in the field of algebraic geometry
MR 0457437 Namikawa, Yukihiko (1980). Toroidal Compactification of Siegel Spaces. Lecture Notes in Mathematics. Vol. 812. doi:10.1007/BFb0091051. ISBN 978-3-540-10021-8
Algebraic_variety
Type of regular Hausdorff space
Hausdorff compactification. Among those Hausdorff compactifications, there is a unique "most general" one, the Stone–Čech compactification β X . {\displaystyle
Tychonoff_space
In algebraic geometry, Nagata's compactification theorem, introduced by Nagata (1962, 1963), implies that every abstract variety can be embedded in a complete
Nagata's compactification theorem
Nagata's_compactification_theorem
Riemannian manifold with SU(n) holonomy
supercharges in a compactification of type IIA supergravity or 2 5 − n {\displaystyle 2^{5-n}} supercharges in a compactification of type I. When fluxes
Calabi–Yau_manifold
Japanese mathematician
known for his work in the field of commutative algebra. Nagata's compactification theorem shows that algebraic varieties can be embedded in complete
Masayoshi_Nagata
Type of topological space in mathematics
cannot be a neighbourhood of any point in Hilbert space. The one-point compactification of the rational numbers Q is compact and therefore locally compact
Locally_compact_space
Type of topological space
In mathematics, a Parovicenko space is a topological space similar to the space of non-isolated points of the Stone–Čech compactification of the integers
Parovicenko_space
Algebraic variety that is a moduli space for principally polarized abelian varieties
particular, a compactification of A2(2) is birationally equivalent to the Segre cubic which is in fact rational. Similarly, a compactification of A2(3) is
Siegel_modular_variety
Jean-Pierre Serre, a French mathematician. Bass–Serre theory Borel-Serre Compactification Grothendieck-Serre Correspondence Serre class Quillen–Suslin theorem
List of things named after Jean-Pierre Serre
List_of_things_named_after_Jean-Pierre_Serre
Mathematical object
with these properties. The 3-sphere is homeomorphic to the one-point compactification of R3. In general, any topological space that is homeomorphic to the
3-sphere
Czech mathematician (1893–1960)
nerve Stone–Čech compactification Tychonoff's theorem O'Connor, John J.; Robertson, Edmund F., "Eduard Čech", MacTutor History of Mathematics Archive, University
Eduard_Čech
Mathematical functions of split-complex numbers
fractional transformations as bijections on the projective line a compactification of D is used. See the section given below. The exponential function
Motor_variable
Research and education institute in Chennai, India
of binary forms in representation theory, the Donaldson-Uhlenbeck compactification in algebraic geometry, stochastic games, inductive algebras of harmonic
Chennai Mathematical Institute
Chennai_Mathematical_Institute
Simple Lie group; the automorphism group of the octonions
In mathematics, G2 is three simple Lie groups (a complex form, a compact real form and a split real form), their Lie algebras g 2 , {\displaystyle {\mathfrak
G2_(mathematics)
Mathematical model of the time dependence of a point in space
orbits and the distinction between different compactifications may be relevant. A category X of mathematical objects has a semigroup G of homomorphisms
Dynamical_system
of mathematics. Alexandrov topology Cantor space Co-kappa topology Cocountable topology Cofinite topology Compact-open topology Compactification Discrete
List of examples in general topology
List_of_examples_in_general_topology
Characterizing property of mathematical constructions
ring, Dedekind–MacNeille completion, product topologies, Stone–Čech compactification, tensor products, inverse limit and direct limit, kernels and cokernels
Universal_property
Line formed by the real numbers
and the resulting end compactification is the extended real number line [−∞, +∞]. There is also the Stone–Čech compactification of the real line, which
Number_line
Book by Lynn Steen
real line Special subsets of the plane One point compactification topology One point compactification of the rationals Hilbert space Fréchet space Hilbert
Counterexamples_in_Topology
In physics and geometry: conjectured relation between pairs of Calabi–Yau manifolds
physics based on string theory, this is accomplished by a process called compactification, in which the extra dimensions are assumed to "close up" on themselves
Mirror symmetry (string theory)
Mirror_symmetry_(string_theory)
topology. Every compact metric space, more generally every one-point compactification of a locally compact metric space, is Eberlein compact. The converse
Eberlein_compactum
Japanese mathematician
received his Ph.D. in mathematics in 1992 after completing his doctoral dissertation, titled "The geometry of the compactification of the Hurwitz scheme
Shinichi_Mochizuki
Duality for locally compact abelian groups
to characterize the Bohr compactification of an arbitrary abelian locally compact topological group. The Bohr compactification B ( G ) {\displaystyle B(G)}
Pontryagin_duality
Study of vector bundles, principal bundles, and fibre bundles
In mathematics, and especially differential geometry and mathematical physics, gauge theory is the general study of connections on vector bundles, principal
Gauge_theory_(mathematics)
Concept in mathematics
William; MacPherson, Robert (January 1994). "A Compactification of Configuration Spaces". Annals of Mathematics. 139 (1): 183. doi:10.2307/2946631. ISSN 0003-486X
Configuration space (mathematics)
Configuration_space_(mathematics)
Class of mathematical expression
\}} is the projectively extended real line, which is a one-point compactification of the real line. Here ∞ {\displaystyle \infty } means an unsigned
Division_by_zero
Danish mathematician and footballer (1887–1951)
than mathematicians. Bohr–Mollerup theorem Bohr compactification Bohr–Favard inequality Danish Mathematical Society List of select Jewish football (association;
Harald_Bohr
Skeletonized version of algebraic geometry
In mathematics, tropical geometry is the study of polynomials and their geometric properties when addition is replaced with minimization and multiplication
Tropical_geometry
Generalization of algebraic variety
affine scheme whose underlying topological space is the Stone–Čech compactification of the positive integers (with the discrete topology). In fact, the
Scheme_(mathematics)
Operation combining two oriented knots
equivalent to R3 with a single point added at infinity (see one-point compactification). A knot is tame if and only if it can be represented as a finite closed
Knot_(mathematics)
Type of topological space
space underlying any profinite group is a Stone space. The Stone–Čech compactification of the natural numbers with the discrete topology, or indeed of any
Stone_space
Indian mathematician
bundles over algebraic varieties, in particular on the Uhlenbeck-Yau compactification of the Moduli Spaces of μ-semistable bundles." He was elected Fellow
Vikraman_Balaji
long as we have no article on Martin boundary, see Compactification (mathematics)#Other compactification theories. Bishop, C. (1991), "A characterization
List of probabilistic proofs of non-probabilistic theorems
List_of_probabilistic_proofs_of_non-probabilistic_theorems
List of concrete topologies and topological spaces
Projectively extended real line Stone–Čech compactification Stone topology Stone–Čech remainder Wallman compactification This lists named topologies of uniform
List_of_topologies
American mathematician and professor emeritus
His research focuses on various areas within mathematics, including topology, Stone-Čech compactification, discrete systems, and Ramsey theory. Neil Hindman
Neil_Hindman
Function that "converges" to periodicity
functions are essentially the same as continuous functions on the Bohr compactification of the reals. The space Sp of Stepanov almost periodic functions (for
Almost_periodic_function
Set of mathematical concepts in quantum gravity
needed for computation. By utilizing compactifications, string theory describes geometric states, where a compactification is a spacetime that looks four-dimensional
Quantum_geometry
Professor of mathematics (born 1969)
his PhD from Harvard University in 1994 with a thesis entitled `A Compactification over the Moduli Space of Stable Curves of the Universal Moduli Space
Rahul_Pandharipande
Objects in eleven-dimensional supergravity
correspondence Phenomenology Phenomenology Cosmology Brane cosmology Landscape Mathematics Geometric Langlands correspondence Mirror symmetry Monstrous moonshine
Supermembranes
completely regular Hausdorff and it contains every point of its Stone–Čech compactification that is real (meaning that the quotient field at that point of the
Realcompact_space
78-dimensional exceptional simple Lie group
In mathematics, E6 is the name of some closely related Lie groups, linear algebraic groups or their Lie algebras e 6 {\displaystyle {\mathfrak {e}}_{6}}
E6_(mathematics)
Concept in mathematical group theory
a complete description, so the alternative complex planes require compactification for complete description of conformal mapping. Nevertheless, the conformal
Conformal_group
smooth elliptic curves, and they also work for the Deligne-Mumford compactification of this moduli stack, in which elliptic curves with nodal singularities
Topological_modular_forms
Taiwanese mathematician
Science in mathematics in 1978. He then earned his Ph.D. in mathematics from Harvard University in 1984. His doctoral thesis, Compactification of the Siegel
Ching-Li_Chai
Topological space in which the closure of every open set is open
space is both extremally disconnected and connected. The Stone–Čech compactification of a discrete space is extremally disconnected. The spectrum of an
Extremally_disconnected_space
uniqueness theorem in geometry Pavel Alexandrov, author of the Alexandroff compactification and the Alexandrov topology Dmitri Anosov, developed Anosov diffeomorphism
List of Russian mathematicians
List_of_Russian_mathematicians
South African mathematician
of European Women in Mathematics. Mathematician Neil Hindman, with whom Strauss wrote a book on the Stone–Čech compactification of topological semigroups
Dona_Strauss
32–33. Gordon, B. Brent (2001) [1994], "Baily–Borel compactification", Encyclopedia of Mathematics, EMS Press Cheeger, Jeff (1983), "Spectral geometry
L²_cohomology
Soviet mathematician (1896–1982)
contributions to set theory and topology. In topology, the Alexandroff compactification and the Alexandrov topology are named after him. Alexandrov attended
Pavel_Alexandrov
geometry) Mumford vanishing theorem (algebraic geometry) Nagata's compactification theorem (algebraic geometry) Noether's theorem on rationality for surfaces
List_of_theorems
American mathematician and professor
Bumsig, Kim; Oh Yong-Geun. A compactification of the space of maps from curves. Transactions of the American Mathematical Society, vol. 366, no. 1, 2014
Andrew_Kresch
Partial differential equations whose solutions are instantons
In physics and mathematics, and especially differential geometry and gauge theory, the Yang–Mills equations are a system of partial differential equations
Yang–Mills_equations
52-dimensional exceptional simple Lie group
In mathematics, F4 is a Lie group and also its Lie algebra f4. It is one of the five exceptional simple Lie groups. F4 has rank 4 and dimension 52. The
F4_(mathematics)
Topology in mathematics
corona set, is the complement βX \ X of the space in its Stone–Čech compactification βX. A topological space is said to be σ-compact if it is the union
Stone–Čech_remainder
Type of topological space
two-point spaces, and a dyadic space is a topological space with a compactification which is a dyadic compactum. However, many authors use the term dyadic
Dyadic_space
German mathematician (born 1972)
lattices and semilattices, and the wonderful compactification. Feichtner earned a diploma in mathematics in 1994 at the Free University of Berlin, and
Eva-Maria_Feichtner
French mathematician (born 1926)
his use of group actions on trees (with Hyman Bass); the Borel–Serre compactification; results on the number of points of curves over finite fields; Galois
Jean-Pierre_Serre
Romanian mathematician and author
University of Paris, where he earned a B.S. degree in 1910 and a Ph.D. in Mathematics in 1916. His doctoral dissertation was written under the direction of
Simion_Stoilow
Chinese mathematician
(MIT) with the thesis Some subvarieties of the De Concini-Procesi compactification under advisor George Lusztig. As a postdoc research fellow, He was
He_Xuhua
Boundary region of asymptotically flat spacetimes in general relativity
{\displaystyle ds^{2}=-dt^{2}+dr^{2}+r^{2}d\Omega ^{2}} . Conformal compactification induces a transformation which preserves angles, but changes the local
Null_infinity
Latvian-American mathematician (1914–1993)
finally proven by Namazi, Souto, and Ohshika in 2010 and 2011. The Bers compactification of Teichmüller space also dates to this period. Over the course of
Lipman_Bers
Function whose actual domain of definition may be smaller than its apparent domain
elements was reinvented many times, in particular, in topology (one-point compactification) and in theoretical computer science." The category of sets and partial
Partial_function
Complement of an open subset
{\displaystyle X} ; the "surrounding space" does not matter here. Stone–Čech compactification, a process that turns a completely regular Hausdorff space into a compact
Closed_set
Theorem of physical impossibility
theorem. Goddard–Thorn theorem Maldacena–Nunez no-go theorem: any compactification of type IIB string theory on an internal compact space with no brane
No-go_theorem
specialized setting, compactifications of topologies correspond to unitizations of algebras. So the one-point compactification corresponds to the minimal
Noncommutative_topology
Topological space in mathematics
long ray, L ∗ , {\displaystyle L^{*},} is obtained as the one-point compactification of L {\displaystyle L} by adjoining an additional element to the right
Long_line_(topology)
American computer scientist and mathematician (born 1938)
Aggregates, nor Stone's Embedding Theorem, nor even the Stone–Čech compactification. (Several students from the civil engineering department got up and
Donald_Knuth
Mathematical concept
Wonderful compactification Homogeneous variety Spherical variety Ash, A.; Mumford, David; Rapoport, M.; Tai, Y. (1975), Smooth compactification of locally
Symmetric_variety
Type of diffeomorphism or homeomorphism of a surface
called the stretch factor or dilatation of f. Thurston constructed a compactification of the Teichmüller space T(S) of a surface S such that the action induced
Pseudo-Anosov_map
British mathematician
algebraic geometry, such as: singularities in the minimal model program; compactification of moduli spaces; the rationality of orbit spaces, including the moduli
Nicholas_Shepherd-Barron
COMPACTIFICATION MATHEMATICS
COMPACTIFICATION MATHEMATICS
COMPACTIFICATION MATHEMATICS
Boy/Male
Scottish
Proud.
Boy/Male
Tamil
Shilpam
Girl/Female
German
Pure; Little and Womanly; Female Version of Charles or Carl
Girl/Female
American, Australian, Spanish
Star
Boy/Male
English
Strong; gifted ruler. Blend of Jer- and Derrick.
Girl/Female
Celtic
Serves God.
Girl/Female
Australian, Celtic, Irish
Radiant Girl; Famous Battle
Surname or Lastname
English
English : variant of Garrett 1.
Girl/Female
American, Australian, British, Danish, English, Finnish, German, Hebrew, Swedish
Consecrated to God; Abbreviation of Elizabeth; God's Promise; God is My Oath
Girl/Female
Hindu, Indian
Queen
COMPACTIFICATION MATHEMATICS
COMPACTIFICATION MATHEMATICS
COMPACTIFICATION MATHEMATICS
COMPACTIFICATION MATHEMATICS
COMPACTIFICATION MATHEMATICS
n.
A preliminary or auxiliary proposition demonstrated or accepted for immediate use in the demonstration of some other proposition, as in mathematics or logic.
a.
Of or pertaining to mathematics; according to mathematics; hence, theoretically precise; accurate; as, mathematical geography; mathematical instruments; mathematical exactness.
n.
One of a school of physicians in Italy, about the middle of the 17th century, who tried to apply the laws of mechanics and mathematics to the human body, and hence were eager student of anatomy; -- opposed to the iatrochemists.
n.
A method of computation; any process of reasoning by the use of symbols; any branch of mathematics that may involve calculation.
n.
One who professed, or publicly teaches, any science or branch of learning; especially, an officer in a university, college, or other seminary, whose business it is to read lectures, or instruct students, in a particular branch of learning; as a professor of theology, of botany, of mathematics, or of political economy.
v. i.
To surpass others in good qualities, laudable actions, or acquirements; to be distinguished by superiority; as, to excel in mathematics, or classics.
n.
The branch of mathematics which studies methods for the calculation of probabilities.
n.
The act of solving, or the state of being solved; the disentanglement of any intricate problem or difficult question; explanation; clearing up; -- used especially in mathematics, either of the process of solving an equation or problem, or the result of the process.
n.
That branch of applied mathematics which teaches the art of determining the area of any portion of the earth's surface, the length and directions of the bounding lines, the contour of the surface, etc., with an accurate delineation of the whole on paper; the act or occupation of making surveys.
n.
That branch of mathematics which treats of the relations of the sides and angles of triangles, which the methods of deducing from certain given parts other required parts, and also of the general relations which exist between the trigonometrical functions of arcs or angles.
n.
One who has made considerable advances in any business, art, science, or branch of learning; an expert; an adept; as, proficient in a trade; a proficient in mathematics, music, etc.
n.
One versed in mathematics.
n.
That science, or class of sciences, which treats of the exact relations existing between quantities or magnitudes, and of the methods by which, in accordance with these relations, quantities sought are deducible from other quantities known or supposed; the science of spatial and quantitative relations.
n.
Learning; especially, mathematics.
a.
Presenting themselves simultaneously and having reciprocal properties; -- frequently used in pure and applied mathematics with reference to two quantities, points, lines, axes, curves, etc.
n.
Mixed mathematics.
n.
That science, or branch of applied mathematics, which treats of the action of forces on bodies.