Baire space

A topological space such that every intersection of a countable collection of open dense sets in the space is also dense.


quasireflexive

(topology) Of a Banach space X: such that the quotient X ′′ / J(X) has finite dimension d. (It is then said to be "quasireflexive of order d".)


Hauptvermutung

From German Hauptvermutung, from haupt- (“main, chief, primary”) + Vermutung (“conjecture”).

(topology) The conjecture, now proven false, that any two triangulations of a triangulable space have a common refinement, a single triangulation that is a subdivision of both of them.


bispace

LL-Q1860 (eng)-Flame, not lame-bispace.wav


pseudo-Anosov

en-adj

pseudo-Anosov (comparative more pseudo-Anosov, superlative most pseudo-Anosov)

more pseudo-Anosov (comparative)
most pseudo-Anosov (superlative)

(topology) Pertaining to the generalization of Anosov diffeomorphism of the torus introduced by William Thurston.


ditopological

Actually a soft ditopological space is a soft set with two independent structures on it - a soft topology and a soft co-topology.

2015, Tugbahan Simsekler Dizman, Alexander Sostak, Saziye Yuksel, “Soft Ditopological Spaces”, in arXiv:


ditopology

(topology) A dichotomous topology; A topology that includes two structures: one describing the properties related to openness of sets, the other describing the properties related to closedness of sets.


disintegrability

Then a topological characterization of the subclasses of extensions satisfying the further properties of full disintegrability and full strong conglomerability is given and their envelopes are studied..

2016, Davide Petturiti, Barbara Vantaggi, “Envelopes of conditional probabilities extending a strategy and a prior probability”, in arXiv:

(topology) The condition of being disintegrable


branch cut

(topology, complex analysis) A curve in the complex plane such that it is possible to define a single analytic branch of a multi-valued function on the plane minus that curve.


topologicality

Switching off magnetism by atomic substitution and ensuring that the electronic structure becomes two-dimensional is sufficient for topologicality to arise in such a system.

2016, J. L. Lado, V. Pardo, “Dirac topological insulator in the d_(z²) manifold of a honeycomb oxide”, in arXiv:

(topology) The condition of being topological


paracomplex

LL-Q1860 (eng)-Flame, not lame-paracomplex.wav

(topology) Pertaining to the category of all metric spaces and all CW-complexes.


aposyndetic

The families of nondegenerate colocally connected or aposyndetic continua in Iⁿ and of at least two-dimensional or decomposable Kelley continua are F#x5F;#x7B;#x5C;sigma#x5C;delta#x7D;-absorbers in the hyperspace C(Iⁿ) for n#x5C;ge 3.

2016, Paweł Krupski, Alicja Samulewicz, “More absorbers in hyperspaces”, in arXiv:

(topology) Having the property that for all points x distinct from a specified point p, there is a compact connected neighborhood of p that does not include x.


halfway model

(topology) An immersion of the sphere S² in R ³, which is the halfway point of a sphere eversion.


Morin surface

(topology) A halfway model of the sphere eversion, featuring fourfold rotational symmetry.


cofinite topology

en-noun

cofinite topology (plural cofinite topologies)

(topology) A topology consisting of the empty set and all cofinite subsets of the underlying set.


underlying set

(topology) A set of "points" upon which a topology is built, thereby yielding a topological space.


trivial topology

trivial topology (plural trivial topologies)

(topology) A topology consisting only of the empty set and the underlying set.


topological property

topological property (plural topological properties)

(topology) A property of a topological space that is invariant under homeomorphisms.


Brouwer fixed-point theorem

the Brouwer fixed-point theorem (canonical)

Named after L. E. J. Brouwer (1881–1966), Dutch mathematician and philosopher.

(topology) A theorem which (in one of its simplest forms) states that a continuous function from an n-dimensional unit cube (in ℝⁿ) to itself has a fixed point.

Brouwer fixed-point theorem
L. E. J. Brouwer


Cauchy problem

After French mathematician Augustin Louis Cauchy.

In this chapter we formulate and in many cases prove results on uniqueness and stability of solutions of the Cauchy problem for general partial differential equations.

2006, Victor Isakov, Inverse Problems for Partial Differential Equations, 2nd edition, Springer, page 41:

To emphasize the similarity between the characteristic method and the probabilistic approach we recall that the method of characteristics allows [one] to reduce the Cauchy problem for a first order PDE to a Cauchy problem for an ODE while the probabilistic approach allows [one] to reduce the Cauchy problem for a second order PDE to a Cauchy problem for an SDE (stochastic differential equation).

2006, S. Albeverio, Ya. Belopolskaya, “Probabilistic Interpretation of the VV-Method for PDE Systems”, in Olga S. Rozanova, editor, Analytical Approaches to Multidimensional Balance Laws, Nova Science Publishers, page 2:

In this monograph we discuss the C#x5C;infty well-posedness of the Cauchy problem for hyperbolic systems.

2014, Tatsuo Nishitani, Hyperbolic Systems with Analytic Coefficients: Well-posedness of the Cauchy Problem, Springer, page v:

mathematical analysis
mathematical analysis

partial differential equation
partial differential equation

(mathematics, mathematical analysis) For a given m-order partial differential equation, the problem of finding a solution function u on ℝⁿ that satisfies the boundary conditions that, for a smooth manifold S⊂ℝⁿ, u(x)=f_0(x) and (∂ᵏu(x))/(∂nᵏ)=f_k(x), ∀x∈S, k=1…m-1, given specified functions f_k defined on, and vector n normal to, the manifold.

problema di Cauchy

class of problem involving simultaneous PDEs


Bolzano-Weierstrass property

the Bolzano-Weierstrass property (canonical)

Named after Bernard Bolzano (1781–1848), a Bohemian mathematician, logician, philosopher, theologian and Catholic priest of Italian descent, and Karl Weierstrass (1815–1897), a German mathematician often cited as the "father of modern analysis".

(topology) The property held by some topological spaces that if a subset of such a space has an infinite quantity of points then the subset has at least one accumulation point.


condensation point

condensation point (plural condensation points)

(topology) (said to be "of" a given subset of a topological space) a point whose every neighborhood contains uncountably many points belonging to the given subset


Lindelöf covering theorem

Named after Ernst Leonard Lindelöf (1870–1946), a Finnish mathematician.

(topology) A theorem which states that for any open covering of some given set in Euclidean n-space there is an open subcovering (of that same set) which is countable.


tangent space

This gives a degenerate norm on the tangent space of Hₙ at the origin, and one can use left translations to define similar norms at the tangent spaces of Hₙ at all other points.

2001, Stephen Semmes, Some Novel Types of Fractal Geometry, Oxford University Press, page 124:

In a Lie group G, because left translation by an element g #x5C;in G is a diffeomorphism that maps a neighborhood of the identity to a neighborhood of g, all the local information about the group is concentrated in a neighborhood of the identity, and the tangent space of the identity assumes a special importance.
Moreover, one can give the tangent space TₑG a Lie bracket [ , ], so that in addition to being a vector space, it becomes a Lie algebra, called the Lie algebra of the Lie group.

2011, Loring W. Tu, An Introduction to Manifolds, 2nd edition, Springer, page 178:

More dramatically, there are situations where one has a sequence of smooth manifolds such that the limit is again a smooth manifold, but with a limiting “tangent space” which has nothing to do with the actual tangent space of the limit!

2014, Peter K. Friz, Martin Hairer, A Course on Rough Paths: With an Introduction to Regularity Structures, Springer, page 7:

(topology, differential topology) An n-dimensional vector space that represents the set of all vectors tangent to given n-dimensional differentiable manifold M at point x.


identification map

identification map (plural identification maps)


quotient map

If the square #x5B;0,1#x5D;#x5C;times#x5B;0,1#x5D; (with the subspace topology) in #x5C;mathbb#x7B;R#x7D;² (with the standard topology) is mapped onto the set #x5B;0,1)#x5C;times#x5B;0,1) with the quotient map f(x,y)#x3D;(#x5C;lfloorx#x5C;rfloor,#x5C;lfloory#x5C;rfloor), then the target space (with its induced quotient topology) is topologically equivalent to a torus.

(topology) A surjective, continuous function from one topological space to another one, such that the latter one's topology has the property that if the inverse image (under the said function) of some subset of it is open in the function's domain, then the subset is open in the target space.


turning number

This program illustrates the Whitney-Graustein theorem, that two closed curves (immersions of the circle) in the plane can be deformed into one another without letting go of the immersion property if and only if their turning numbers are the same.

1989, Wayne Sewell, Weaving a Program: Literate Programming in WEB, page 516:

Moreover, l gives the turning number of the k-gon one sees in the horizontal symmetry plane when the axes of the ends are deleted.[…]Moreover, every even k ≥ 30 except for 32, 36, 40, 44, 48 52 an 56 gives an example, while for every even k > 102 there is more than one example since different turning numbers are possible.

1998, Karsten Groβe-Brauckmann, Robert B. Kusner, John M. Sullivan, “Constant Mean Curvature Surfaces with Cylindrical Ends”, in Hans-Christian Hege, Konrad Polthier, editors, Mathematical Visualization: Algorithms, Applications and Numerics, Springer, page 114:

2002, Journal of Physics: Mathematical and general, page 6187,
For any oriented curve β(τ),τ∈I,I=[0,Λ] of class C¹ (τ is the arclength parametrization) lying in an oriented Euclidean plane E, the turning number 𝒯_(n_β) may be defined as
𝒯_(n_β)=1/(2π)∫₀^Λκ_β(τ)dτ
where κ_β(τ) is the signed curvature of β.

(geometry, topology, mathematical analysis) A version of winding number in which the number of rotations is counted with respect to the tangent of the curve rather than a fixed point.


Whitehead link

(topology) One of the most basic interlinkages of two loops; a link with one ring passing through the two lobes of a singly twisted loop.


Hopf link

From Hopf + link. From being an interlinkage named after Heinz Hopf.

(topology) The simplest interlinkage of two loops; two interlinked rings.


Jordan curve theorem

There is a vast generalization of the Jordan curve theorem to higher dimensions.

1995, William Fulton, Algebraic Topology: A First Course, Springer, page 343:

The Jordan curve theorem asserts that a simple closed curve in the complex plane divides the plane into exactly two connected components, one bounded (the "inside") and the other unbounded (the "outside" ).

2001, Theodore Gamelin, Complex Analysis, Springer, page 249:

The principal reason that orbit behavior in the plane is more limited than in higher dimensions is the Jordan curve theorem. Local sections will be the primary tool used to exploit the Jordan curve theorem.

2004, Nelson G. Markley, Principles of Differential Equations, Wiley, page 193:

(topology) The theorem that states that a simple closed curve (Jordan curve) divides the plane into precisely two distinct areas.


sober space

For a reader more interested in function spaces than in functors, this concludes the description of content except to add that the classification of coadjoint G’s by sets B bearing a topological topology relativizes to T₀ spaces. For other readers: and trivially to sober spaces.

1975, Mathematica Scandinavica, Societates Mathematicae, page 318:

1983, Houston Journal of Mathematics, Volume 9, University of Houston, page 192,
In the Hofmann and Lawson paper, it is proved that the topological space Spec(L) is a locally quasicompact sober space […] .

Thus sober spaces are necessarily T₀.

2002, P. T. Johnstone, Sketches of an Elephant: A Topos Theory Compendium, volume 2, Oxford University Press, page 492:

How rich are sober and L-sober spaces, are there important examples? It is well-known [25] that Hausdorff spaces, and hence compact T₀ (and so finite T₀ spaces), are sober spaces in the traditional setting. Furthermore, the soberification of a space—the spectrum of the topology of a space—is always sober; and if the original space is not Hausdorff, then its soberification is a sober space which is not Hausdorff. So there are many non-Hausdorff sober spaces as well.

2003, A. Pultr, S. E. Rodabaugh, “Chapter 6: Lattice-Valued Frames, Functor Categories, And Classes of Sober Spaces”, in Stephen Ernest Rodabaugh, Erich Peter Klement, editors, Topological and Algebraic Structures in Fuzzy Sets: A Handbook of Recent Developments in the Mathematics of Fuzzy Sets, Springer, page 155:

When X is a noetherian sober space the construction of loc. cit. can be extended to an arbitrary lower-semicontinuous function for the constructible topology.

2004, Ofer Gabber, “Notes on some t-structures”, in Alan Adolphson, Francesco Baldassarri, Pierre Berthelot, Nicholas Katz, François Loeser, editors, Geometric Aspects of Dwork Theory, volume 1, Walter de Gruyter, page 711:

(topology) A topological space of which every join-irreducible closed subset is the closure of exactly one point of the space.


Sierpinski space

Sierpinski space (plural Sierpinski spaces)

Named in honour of Wacław Sierpiński (1882–1969), a Polish mathematician.

Alexandrov-discrete space
Alexandrov-discrete space

(topology) A two-point Alexandrov-discrete space.


generic property

generic property (plural generic properties)

1989, Thomas S. Parker, Leon O. Chua, Appendix C: Differential Topology, Practical Numerical Algorithms for Chaotic Systems, Springer, Softcover reprint, page 314,
A property P that refers to members of a set Y is a generic property if the subset of Y whose members exhibit property P contains a dense open subset of Y.
Remark: If P₁ and P₂ are generic properties of a set Y, then so is the property "P₁ and P₂."

2001, Constantin I. Chueshov (translator), Igor Chueshov, Introduction to the Theory of Infinite-dimensional Dissipative Systems, [1999, Russian edition] Acta Scientific Publishing House, page 117,
It should be noted that if a property of a dynamical system holds for the parameters from an open and dense set in the corresponding space, then it is frequently said that this property is a generic property.
However, it should be kept in mind that the generic property is not the one that holds almost always. […] Therefore, it should be remembered that generic properties are quite frequently encountered and stay stable during small perturbations of the properties of a system.

We now formulate the first generic property concerning the singularities of a Hamiltonian function.

2010, H. W. Broer, F. Takens, “Chapter 1: Preliminaries of Dynamical Systems Theory”, in B. Hasselblatt, H. W. Broer, F. Takens, editors, Handbook of Dynamical Systems, volume 3, Elsevier (North-Holland), page 24:

(topology, algebraic geometry) A property that is true in some dense open subset of a given set.


vector bundle

Let E and F be two vector bundles on a compact algebraic set X.

1995, Mauro Beltrametti, Andrew John Sommese, The Adjunction Theory of Complex Projective Varieties, Walter de Gruyter, page 8:

2012, Igor Burban, Bernd Kreussler, Vector Bundles on Degenerations of Elliptic Curves and Yang-Baxter Equations, American Mathematical Society, page 5,
This solutions actually depend on the choice of a trivialisation ξ of the universal family 𝒫(n,d) of stable vector bundles of rank n and degree d.

Definition 6.1.5 A non-zero vector bundle #x5C;mathcal#x7B;F#x7D; on X is said to be decomposable if it can be written as #x5C;mathcal#x7B;F#x7D;#x5F;1#x5C;oplus#x5C;mathcal#x7B;F#x7D;#x5F;2 with #x5C;mathcal#x7B;F#x7D;#x5F;1 and #x5C;mathcal#x7B;F#x7D;#x5F;2 non-zero vector bundles. Otherwise it is said to be indecomposable.

2017, David J. Benson, Representations of Elementary Abelian p-Groups and Vector Bundles, Cambridge University Press, page 130:

(topology, category theory) A fiber bundle for which the fiber is a vector space.


tangent bundle

(topology) A fiber bundle for which the base space is a differentiable manifold and each fiber over a point of that manifold is the tangent space of that point.


differential structure

differential structure (plural differential structures)

First let (M,#x5C;tau) be a topological space. The sheaf #x5C;mathfrak#x7B;G#x7D; of real continuous functions on (M,#x5C;tau) is said to be a differential structure on M if for any open set U#x5C;in#x5C;tau, any functions f#x5F;i#x5C;in#x5C;mathfrak#x7B;G#x7D;(U), and any w#x5C;inC#x5C;infty(#x5C;mathbb#x7B;R#x7D;ⁿ), the superposition w#x5C;circ(f#x5F;1,#x5C;dotsf#x5F;n)#x5C;in#x5C;mathfrak#x7B;G#x7D;(U).

2002, R.W. Carroll, Calculus Revisited, Springer, pages 12–37:

It is important to emphasise that, among the various choices for #x5C;lambdaⁱ#x5F;#x7B;jk#x7D; and A#x7B;i#x5C;a#x7D;#x5F;#x7B;#x5C;j#x5C;b#x7D;, some are intrinsic to the differential structure of the manifold #x5C;mathcal#x7B;M#x7D;. In other words, among all the operators of D-differentiation, some arise from the differential structure of #x5C;mathcal#x7B;M#x7D;.[…]On the other hand, there exist operators of D-differentiation that do not follow from the differential structure of #x5C;mathcal#x7B;M#x7D;.

2002, Donal J. Hurley, Michael A. Vandyck, Topics in Differential Geometry: A New Approach Using D-Differentiation, Springer (with Praxis Publishing), page 29:

This chapter is concerned with differentiable measures on general measurable spaces and on measurable spaces equipped with certain differential structures enabling us to consider differentiations along vector fields.

2010, Vladimir Igorevich Bogachev, Differentiable Measures and the Malliavin Calculus, American Mathematical Society, page 369:

Given a Hausdorff topological space M with differential structures #x5C;mathcal#x7B;A#x7D;#x5F;1 and #x5C;mathcal#x7B;A#x7D;#x5F;2 (these being maximal smooth atlases), we say that #x5C;mathcal#x7B;A#x7D;#x5F;1 and #x5C;mathcal#x7B;A#x7D;#x5F;1 are equivalent if there is a diffeomorphism #x5C;phi#x3A;(M,#x5C;mathcal#x7B;A#x7D;#x5F;1)#x5C;rightarrow(M,#x5C;mathcal#x7B;A#x7D;#x5F;2) from M with the first differential structure to M with the second differential structure. Note that #x5C;phi need not be the identity function.

2015, Stephen Bruce Sontz, Principal Bundles: The Classical Case, Springer, page 12:

(topology) A structure defined for a (topological) manifold so that it supports differentiation of functions defined on it.


disjoint union topology

disjoint union topology (plural disjoint union topologies)

The reader will recall that the morphism and object sets of A'#x3D;#x5C;coprod#x5F;#x7B;p,q#x5C;inP#x7D;A#x5F;c(q,p) were not necessarily given the disjoint union topologies.

1978, Maria Louise Shea Terrell, Topological 2-categories and Principal Topological Categories, University of Virginia, page 19:

Given any indexed collection of topological spaces #x5C;left#x5C;#x7B;X#x5F;#x5C;alpha#x5C;right#x5C;#x7D;#x5F;#x7B;#x5C;alpha#x5C;inA#x7D;, we define the disjoint union topology on #x5C;textstyle#x5C;coprod#x5F;#x7B;#x5C;alpha#x5C;inA#x7D;X#x5F;#x5C;alpha by declaring a subset of #x5C;textstyle#x5C;coprod#x5F;#x7B;#x5C;alpha#x5C;inA#x7D;X#x5F;#x5C;alpha to be open if and only if its intersection with each X#x5F;#x5C;alpha is open in X#x5F;#x5C;alpha.

2003, John M. Lee, Introduction to Smooth Manifolds, Springer, page 548:

And the set Gⁿ(#x5C;mathbb#x7B;F#x7D;) of all sub vector spaces of #x5C;mathbb#x7B;F#x7D;ⁿ, i.e. the disjoint union of the Gⁿ#x5F;k(#x5C;mathbb#x7B;F#x7D;), gets the disjoint union topology.

2005, Friedrich Ischebeck, Ravi A. Rao, Ideals and Reality: Projective Modules and Number of Generators of Ideals, Springer, page 118:

(topology) A topology which is applicable to the disjoint union of a given set of topological spaces and is the largest (most inclusive) topology that preserves the continuity of each contributing space as represented in the union.


Alexandrov topology

Alexandrov topology (plural Alexandrov topologies)

Along with the closely related concept of Alexandrov-discrete space, after Russian mathematician Pavel Sergeyevich Alexandrov.

4.24. THEOREM. The following three restrictions on a space-time M are equivalent:
(a) M is strongly causal;
(b) the Alexandrov topology agrees with the manifold topology;
(c) the Alexandrov topology is Hausdorff.

1972, Roger Penrose, Techniques of Differential Topology in Relativity, SIAM, page 34:

In Minkowski space-time there is, indeed, a topology definable solely in terms of the causal connectivity among events, the Alexandrov topology, which is provably equivalent to the ordinary manifold topology.

1990, Palle Yourgrau, Demonstratives, Oxford University Press, page 252:

2016, Piero Pagliani, Covering Rough Sets and Formal Topology — A Uniform Approach Through Intensional and Extensional Operators, James F. Peters, Andrzej Skowron (editors), Transactions on Rough Sets XX, Springer, LNCS 10020, page 134,
(6) S_0,2(P(R_C)) is the family of open sets of the Alexandrov topology induced by R_C and S(P(R_C)) is the family of closed sets of the Alexandrov topology induced by R^⌣_C;

(topology) A topology in which the intersection of any family of open sets is an open set.

Alexandrov topology
Pavel Sergeyevich Alexandrov


topologization

Many of the topological concepts that will be defined later can be incorporated into methods of topologization in the same general way.

1963, Donald Bushaw, Elements of General Topology, Wiley, page 29:

1968, A. F. Mutylin, An Example of Nontrivial Topologization of the Field of Rational Numbers: Complete Locally Bounded Fields, V. P. Kompaniec (editor), Fourteen Papers on Algebra, Topology, Algebraic and Differential Geometry, American Mathematical Society Translations, Series 2, Volume 73, page 164,
The p-adic topologization also does not coincide with ours and is not stronger than it since any Uₙ contains Pₖ which is an integer and which is relatively prime with p.

The problem on the non-discrete Hausdorff topologization of rings was investigated in a series of works.

2005, Vladimir Arnautov, Kirill Filippov, “On Group Topologies on an Abelian Group Preceding One Another”, in Svetlana Cojocaru, Gerhard Pfister, Victor Ufnarovski, editors, Computational Commutative and Non-commutative Algebraic Geometry, IOS Press, page 251:

(topology) The act or process of topologizing; the result of a process of topologizing, especially the topology so produced.


profinite group

profinite group (plural profinite groups)

This proposition shows that to have a representation with big image of a profinite group, for example, a Galois group, we need to take a profinite topological ring A as a coefficient ring, like the p-adic integer ring #x5C;Z#x5F;p.

2000, Haruzo Hida, Modular Forms and Galois Cohomology, Cambridge University Press, page 28:

We have already pointed out that a profinite group G of an infinite rank m is free of rank m if (and only if) G is projective and every finite split embedding problem for G with a nontrivial kernel has m solutions (Proposition 9.4.7). Dropping the condition on G to be projective leads to the notion of a "quasi-free profinite group" (Section 10.6).

2011, Moshe Jarden, Algebraic Patching, Springer, page 207:

2014, David Barnes, Constanze Roitzheim, Rational equivariant rigidity, Christian Ausoni, Kathryn Hess, Brenda Johnson, Wolfgang Lück, Jérôme Scherer, An Alpine Expedition through Algebraic Topology, American Mathematical Society, page 14,
Recall that a profinite group is an inverse limit of an inverse system of finite groups, with the p-adic numbers being the canonical example. Any finite group is, of course, profinite, but when we talk of a profinite group we assume that the group is infinite.

(topology, category theory) A topological group that is isomorphic to the inverse limit of some inverse system of discrete finite groups; equivalently, a topological group that is also a Stone space.

groupe profini

topological group isomorphic to the inverse limit of an inverse system of discrete finite groups


geometric analysis

Papers are submitted for publication by sending three hard (printed) copies to The Journal of Geometric Analysis, Department of Mathematics, Washington University,[…].

1999, The Journal of Geometric Analysis, volume 9, CRC Press, page 273:

2013, Mohammad Ghomi, Junfang Li, John McCuan, Vladimir Oliker, Fernando Schwartz, Gilbert Weindtein (editors) Geometric Analysis, Mathematical Relativity, and Nonlinear Partial Differential Equations, American Mathematical Society.

In 2012, thematic programs on Geometric Analysis and Spectral Theory and on Moduli spaces, Extremality and Global Invariants were held at the CRM. There were several workshops on cutting-edge topics in geometric analysis and spectral theory, notably on the Geometry of Eigenvalues and Eigenfunctions, on Manifolds of Metrics and Probabilistic Methods in Geometry and Analysis and on Spectral invariants on non-compact and singular spaces.

2014, Pierre Albin, Dmitry Jakobson, Frédéric Rochon, “Preface”, in Geometric and Spectral Analysis, American Mathematical Society, page vii:

differential geometry
differential geometry

(mathematical analysis, geometry, topology) A branch of mathematics in which techniques from differential geometry are used in the analysis and solution of partial differential equations, and vice versa.


lake of Wada

Calque of Japanese 和田の湖. Introduced by Kunizo Yoneyama (in 1917), who credited the discovery to Takeo Wada.

(mathematics) Any of three disjoint connected open sets of the plane or open unit square with the counterintuitive property that they all have the same boundary.

Kunizo Yoneyama
Lakes of Wada
Takeo Wada


Chern class

(mathematics, topology, geometry) Any of a group of characteristic classes associated with complex vector bundles, with applications in physics, string theory, etc.


mathematical space

mathematical space (plural mathematical spaces)

(topology) A set (a "universe") that consists of selected mathematical objects that are treated as points, and selected relationships between them.

mathematischer Raum

a set that consists of selected mathematical object


metacompact

(topology) Of a topological space: such that every open cover has a point finite open refinement. That is, given any open cover of the topological space, there is a refinement which is again an open cover with the property that every point is contained only in finitely many sets of the refining cover.


H-space

The H was suggested by Jean-Pierre Serre in recognition of the influence exerted on the subject by Heinz Hopf.

(topology) A topological space X (generally assumed to be connected) together with a continuous map μ : X × X → X with an identity element e such that μ(e, x) = μ(x, e) = x for all x in X.


locally compact group

locally compact group (plural locally compact groups)

Indeed, if there is one property of locally compact groups more responsible than any other for the rich positive content of their representation theory, it is their possession of left-invariant and right-invariant (Haar) measures. The connection between the representation theory of a locally compact group and that of general Banach algebras proceeds directly from Lebesgue integration with respect to Haar measure on the group.

1988, J. M. G. Fell, R. S. Doran, Representations of *-Algebras, Locally Compact Groups, and Banach *-Algebraic Bundles, Volume 1, Academic Press, page 63:

A locally compact group G is called Lie projective if it is the projective limit #x5C;textstyle#x5C;underset#x7B;#x5C;leftarrow#x7D;#x7B;#x5C;lim#x7D;#x5F;#x7B;#x5C;alpha#x5C;in#x5C;mathbb#x7B;A

2012, H. Heyer, Probability Measures on Locally Compact Groups, Springer, page 12:

Let G be a locally compact group and H a closed subgroup of G, and suppose that #x5C;pi and #x5C;tau are irreducible representations of G and H, respectively.

2013, Eberhard Kaniuth, Keith F. Taylor, Induced Representations of Locally Compact Groups, Cambridge University Press, page 269:

(topology) A topological group whose underlying topology is both locally compact and Hausdorff.

lokalkompakte Gruppe

topological group whose topology is locally compact and Hausdorff


algebraic analysis

The area of study in algebraic geometry began as a research programme, instigated in 1959 by Japanese mathematician Mikio Sato.

1986, Goro Kato (translator), Masaki Kashiwara, Takahiro Kawai, Tatsuo Kimura, Foundations of Algebraic Analysis, Princeton Universty Press, https://books.google.com.au/books?id=qYUrDgAAQBAJ&printsec=frontcover&dq=%22algebraic+analysis%22&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwiK5MSlvsLsAhWXbn0KHbPuBPQQ6AEwAnoECAYQAg#v=onepage&q=%22algebraic%20analysis%22&f=false.

In their classical work [123], Kashiwara, Kawai, and Kimura pointedly observe that the Japanese algebraic analysis is really the algebraic analysis in the tradition of Euler; we may add that what we mean by algebraic microlocal analysis is the successful attempt to adapt the methods of abstract algebraic geometry to the non-commutative setting in which the base ring is now the ring of variable coefficients^([sic]) partial differential operators.

1999, Goro Kato, Daniele C. Struppa, Fundamentals of Algebraic Microlocal Analysis, Marcel Dekker, page v:

2008, Yoshitsugu Takei, The work of T. Kawai on exact WKB analysis, T. Aoki, H. Majima, Y. Takei, N. Tose (editors), Algebraic Analysis of Differential Equations, Springer, page 19,
Thus Kawai has been engaged mainly in the research of exact WKB analysis or algebraic analysis of singular perturbation theory since 1989.

(algebraic geometry, algebraic topology) The study of systems of linear partial differential equations, using techniques from sheaf theory and complex analysis to examine functions and certain generalisations of functions (such as hyperfunctions and microfunctions).


final topology

final topology (plural final topologies)

topological space
topological space#English

(topology) Given a set X, a family of topological spaces (M#x5F;i,T#x5F;i)#x5F;#x7B;i#x5C;inI#x7D; and a family of functions f#x5F;i#x3A;M#x5F;i#x5C;toX, the final topology is the finest topology on X such that all f#x5F;i are continuous.


initial topology

initial topology (plural initial topologies)

(topology) Given a set X, a family of topological spaces (M#x5F;i,T#x5F;i)#x5F;#x7B;i#x5C;inI#x7D; and a family of functions f#x5F;i#x3A;X#x5C;toM#x5F;i, the initial topology is the coarsest topology on X such that all f#x5F;i are continuous.


closed loop

(topology) A path whose initial point is equal to its terminal point.


chain group

kth chain group (canonical)
chain groups (plural)

(algebraic topology) A free abelian group generated by all the k-dimensional oriented simplices of a given simplicial complex.


homology group

kth homology group (plural homology groups)

kth homology group (canonical)
homology groups (plural)

(algebraic topology) The quotient group of the kth cycle group modulo the kth boundary group (derived from the chain complex of, e.g., a given simplicial complex).


boundary group

kth boundary group (plural boundary groups)

kth boundary group (canonical)
boundary groups (plural)

boundary homomorphism
boundary homomorphism

(algebraic topology) The image of the (k + 1)ᵗʰ boundary homomorphism (for a given chain complex).


cycle group

kth cycle group (canonical)
cycle groups (plural)

(algebraic topology) The kernel of the kᵗʰ boundary homomorphism (of a given chain complex).


boundary homomorphism

kth boundary homomorphism (plural boundary homomorphisms)

kth boundary homomorphism (canonical)
boundary homomorphisms (plural)

(algebraic topology, in a simplicial chain complex) A group homomorphism on the group of simplicial k-chains induced by mapping each oriented k-simplex to a particular formal sum of its faces.

(algebraic topology, in greater generality) A homomorphism that operates on the kth element in a chain complex.


Alexander horned sphere

Alexander horned sphere (plural Alexander horned spheres)

(topology) A pathological object, a certain embedding of a sphere in three-dimensional Euclidean space, obtained by various modifications of a torus.


exact cover

A finite set #x5C;#x7B;S(#x5C;alpha#x5F;i,#x5C;beta#x5F;i)#x5C;vert 1#x5C;lei#x5C;lem#x5C;#x7D; is called an (eventual) exact cover if every (sufficiently large) positive integer occurs in exactly one S(#x5C;alpha#x5F;i,#x5C;beta#x5F;i). If #x5C;#x7B;S(#x5C;alpha#x5F;i,#x5C;beta#x5F;i)#x5C;#x7D;#x5F;#x7B;i#x3D;1#x7D;ᵐ is an eventual exact cover, then #x5C;textstyle#x5C;sum#x5F;#x7B;i#x3D;1#x7D;ᵐ#x7B;#x5C;alpha#x5F;i#x7B;-1

2000, R. Tijdeman, “Exact covers of balanced sequences and Fraenkel's conjecture”, in F. Halter-Koch, Robert F. Tichy, editors, Algebraic Number Theory and Diophantine Analysis: Proceedings of the International Conference, Walter de Gruyter, page 468:

2011, R. Lu, S. Liu, J. Zhang, Searching for Doubly Self-orthogonal Latin Squares, Jimmy Lee (editor), Principles and Practice of Constraint Programming: 17th International Conference CP 2011, Proceedings, Springer, LNCS 6876, page 542,
It is straightforward to use clique algorithms to construct a (partial) solution of a given combinatorial problem which is represented as a set system. If the solution of the combinatorial problem corresponds to the exact cover of the set system, a substantially more efficient algorithm can be utilized because of this property.

(topology, combinatorics) Given a collection S of subsets of a set X, a subcollection S^* of S such that each element of X is contained in exactly one subset in S^*.

exakte Überdeckung

Collection of subsets such that each element of the original set is contained in exactly one subset


merotopy

An affirmative answer was given by Simon in a recent paper on natural merotopies.

1980, Topology (issue 23, part 1, page 608)

That μ_A satisfies the axioms of a merotopy is easily seen.

2004, Categorical Structures and Their Applications: Proceedings of the North-West European Category Seminar, Berlin, Germany, 28-29 March 2003, page 63:

(topology) The set of all subsets of a topological space such that, for some element in that space every neighbourhood of that element contains a member of the subset.


Brunnian braid

(topology) A braid that becomes trivial upon removal of any one of its strings.


spatial locale

(topology) A locale which originates from a topological space.


Stone-Čech compactification

Stone-Čech compactification (countable and uncountable, plural Stone-Čech compactifications)

Named after Marshall Stone and Eduard Čech, who wrote on the topic in 1937.

A technique for constructing a universal map from a topological space to a compact Hausdorff space.

The space obtained by using this technique.


Čech-Stone compactification

Čech-Stone compactification (countable and uncountable, plural Čech-Stone compactifications)

Stone-Čech compactification
Stone-Čech compactification#English

Alternative form of Stone-Čech compactification


connecting homomorphism

connecting homomorphism (plural connecting homomorphisms)

(homological algebra) The homomorphism constructed by the snake lemma, leading to a particular exact sequence of kernels and cokernels.

(algebraic topology) Such a homomorphism in the context of exact sequences of chain complexes, leading to a long exact sequence of homology groups.


skeletoid

By proposition 6.5.4, X is a skeletoid and hence an absorber by theorem 6.5.1.

1988, J. van Mill, Infinite-Dimensional Topology: Prerequisites and Introduction, page 285:

Skeletoids and absorbers play an important role in infinite-dimensional topology. The existence and uniqueness of skeletoids (and absorbers) with respect to certain collections allow ^([sic]) one to study completely metrizable infinite-dimensional manifolds by using the technique of compact manifolds, incomplete manifolds by those of complete ones and so on.

2020, Howard Cook, Continua: With the Houston Problem Book, page 70:

(topology) A structure that captures the essential topological information of a space.


sequentially compact

(topology, of a topological space) Such that every sequence in the space has a convergent subsequence. In a metric space, this is equivalent to being compact (sense 4.1).

jonokompakti

topology: every sequence in the space has a convergent subsequence


time-one map

In [5], Brin and Pesin investigated the dynamical properties of diffeomorphisms like the time-one map of an Anosov flow, which they called "partially hyperbolic."

1995, Anne Marie Wilkinson, Stable Ergodicity of the Time-one Map of a Geodesic Flow, page 12:

(mathematics, topology) A type of diffeomorphism that represents the evolution of a dynamical system over a unit time interval, where each point in the system's phase space is mapped to another point after one unit of time has passed.


Eötvös effect

(physics) The change in measured Earth's gravity caused by the change in centrifugal acceleration resulting from eastbound or westbound velocity.


Urysohn's lemma

Named after the mathematician Pavel Samuilovich Urysohn.

(topology) A lemma stating that a topological space is normal if and only if any two disjoint closed subsets can be separated by a continuous function.


bouquet of spheres

bouquet of spheres (plural bouquets of spheres)

(topology) A wedge product of two or more spheres joined at a distinguished point, usually the equator.