قائمة مواضيع الطوبولوجيا

topology glossary
list of geometric topology topics
list of algebraic topology topics

Basic concepts

فضاء طوبولوجي Topological space
خاصية طوبولوجية Topological property
مجموعة مفتوحة
مجموعة مغلقة closed set
- Clopen set
- Closure (topology)
- Boundary (topology), boundary
- Dense (topology)
- G-delta set, F-sigma set
- closeness (mathematics)
- neighbourhood (mathematics)
Continuity (topology)
- هوميومورفية Homeomorphism
- هوميومورفية محلية Local homeomorphism
- Open and closed maps
- Germ (mathematics)
Base (topology), subbase
Open cover
Atlas (topology)

Limits

Limit point
Net (topology)
Filter (topology)
Ultrafilter

Topological properties

Baire category theorem
- Nowhere dense
- Baire space
- لعبة باناخ مازور Banach-Mazur game
- Co-meagre

Compactness and countability

Compact space
- Relatively compact subspace
- مبرهنة هاينه وبوريل Heine-Borel theorem
- مبرهنة تيخونوفس Tychonoff's theorem
- خاصية التقاطع المحدود Finite intersection property
- Compactification
- Measure of non-compactness
Paracompact space
Locally compact space
Compactly generated space
حدسية قابلية العد Axiom of countability
First-countable space
Second-countable space
Separable space
فضاء ليندلوف Lindelöf space
Sigma-compactness

Connectedness

Connected space

Separation axioms

T0 space
T1 space
فضاء هاوس دورف Hausdorff space
- Completely Hausdorff space
Regular space
Tychonoff space
Normal space
Urysohn's lemma
Tietze extension theorem
Paracompact
Separated sets

Topological constructions

direct sum and the dual construction product
subspace and the dual construction quotient

Examples

See also: List of examples in general topology.

Discrete space
- Locally constant function
Trivial topology
Cofinite topology
Finer topology
Product topology
- Restricted product
Quotient space
Unit interval
Continuum (mathematics)
Extended real number line
Long line (topology)
Sierpinski space
Cantor set, Cantor space
Space-filling curve
Topologist's sine curve
Uniform norm
Weak topology
Strong topology
Hilbert cube
Lower limit topology
Sorgenfrey plane
Real tree
Compact-open topology
Zariski topology
Kuratowski closure axioms
Unicoherent
Solenoid (mathematics)

Uniform spaces

Uniform continuity
Lipschitz continuity
Uniform isomorphism
Uniform property
Uniformly connected space

Metric spaces

Metric topology
Manhattan distance
Ultrametric space
- P-adic numbers, p-adic analysis
Open ball
Bounded subset
Pointwise convergence
Metrization theorems
Complete space
- Cauchy sequence
- Banach fixed point theorem
Polish space
Hausdorff distance
Intrinsic metric
Category of metric spaces

Topology and order theory

Stone duality
- Stone's representation theorem for Boolean algebras
Specialization (pre)order
Sober space
Spectral space
Alexandrov topology
Upper topology
Scott topology
- Scott continuity
Lawson topology

Descriptive set theory

Dimension theory

See also main article dimension

Inductive dimension
Lebesgue covering dimension
Lebesgue's number lemma

Topological algebra

Topological group
Topological abelian group
Topological ring
Topological vector space
Properly discontinuous
Sheaf space

Combinatorial topology

Polytope
Simplex
Simplicial complex
CW complex
Manifold
Triangulation
Barycentric subdivision
Sperner's lemma
Simplicial approximation theorem
Nerve of an open covering

Foundations of algebraic topology

Simply connected
Semi-locally simply connected
Path (topology)
Homotopy
Homotopy lifting property
Pointed space
Wedge sum
smash product
Cone (topology)
Adjunction space

Differential geometry of curves

List of curve topics
Frenet-Serret formulas
Curves in differential geometry
Line element
Curvature
Radius of curvature
Osculating circle
Curve

Differential geometry of surfaces

Ruled surface
Conical surface
Mean curvature
Theorema egregium
Gauss-Bonnet theorem

Calculus on manifolds

See also multivariable calculus, list of multivariable calculus topics

Manifold
Tangent vector
Tangent space
Tangent bundle
Cotangent space
Cotangent bundle
Vector bundle
Vector field
Tensor field
Differential form
Exterior derivative
مبرهنة بيرون فروبانيوس Frobenius theorem
Contact (mathematics)

Differential topology

Diffeomorphism
- Large diffeomorphism
Orientability
Whitney embedding theorem
Critical value
Saddle point
Morse theory
Lie derivative
Hairy ball theorem
Poincaré-Hopf theorem
Stokes' theorem
De Rham cohomology
Smale's paradox

Fiber bundles

Fiber bundle
Principal bundle
- Frame bundle
- Hopf bundle
Associated bundle
Vector bundle
- Tangent bundle
- Cotangent bundle
- Line bundle
Jet bundle

Riemannian geometry

List of coordinate charts
Metric tensor
Riemannian manifold
Non-Euclidean geometry
- Angle excess
- Sphere-world
Hodge star operator
Killing vector field
Geodesic
- Prime geodesic
Geodesic flow
Exponential map
Injectivity radius
Jacobi field
Uniformization theorem
Myers theorem
Symmetric space
Hypersurface
Nash embedding theorem
Gromov's compactness theorem
Hsiang-Lawson's conjecture
Riemannian submanifold

Variant structures

Intrinsic metric
Pseudo-Riemannian manifold
Sub-Riemannian manifold
General relativity
Holonomy, local holonomy
G-structure
Complex manifold,
Almost complex manifold
Kähler manifold
- Calabi-Yau spaces
Hyperkähler manifold
Symplectic topology
Contact geometry
G2 manifold
Foliation
Integrability conditions for differential systems
Finsler geometry
Information geometry

Curvature

Main article curvature of Riemannian manifolds

Theorema Egregium
Gauss-Bonnet theorem
Gauss map
Second fundamental form
Curvature vector
Curvature form
Curvature tensor
Geodesic curvature
Scalar curvature
Sectional curvature
Ricci curvature, Ricci flat
Weyl curvature
Ricci flow
Minimal surface
Connection
- Connection form
  - Chern-Gauss-Bonnet theorem
  - Chern-Weil homomorphism
- Cartan connection
  - Cartan connection applications
  - Vierbein
  - Einstein-Cartan theory
- Covariant derivative
- Levi-Civita connection
- Moving frame
- Parallel transport
- Torsion (differential geometry)

Other

Envelope (mathematics)
Bäcklund transform

Classical topics in projective geometry

Affine space
Projective space
Projective line, cross-ratio
Projective plane
- Line at infinity
- Complex projective plane
Complex projective space
Plane at infinity, hyperplane at infinity
Projective frame
Projective transformation
Fundamental theorem of projective geometry
Duality (projective geometry)
Real projective plane
Real projective space
Segre embedding, multi-way projective space
Rational normal curve

Algebraic curves

Conics, Pascal's theorem, Brianchon's theorem
Twisted cubic
Elliptic curve, cubic curve
- Elliptic function, Jacobi's elliptic functions, Weierstrass's elliptic functions
- Elliptic integral
- Complex multiplication
- Weil pairing
Hyperelliptic curve
Klein quartic
modular curve
- modular equation
- modular function
- modular group
- supersingular prime
Fermat curve
Bézout's theorem
Brill-Noether theory
Genus (mathematics)
Riemann surface
Riemann-Hurwitz formula
Riemann-Roch theorem
Abelian integral
Differential of the first kind
Jacobian variety
- Generalized Jacobian
Hurwitz's automorphisms theorem
Clifford's theorem
Gonality of an algebraic curve
Weil's reciprocity law
Goppa code

Algebraic surfaces

Enriques-Kodaira classification
List of algebraic surfaces
Ruled surface
Cubic surface
Veronese surface
Del Pezzo surface
Rational surface
Enriques surface
K3 surface
Hodge index theorem
Elliptic surface
Surface of general type
Zariski surface

Algebraic geometry: classical approach

Algebraic variety
- Hypersurface
- Quadric
- Dimension of an algebraic variety
- Hilbert's Nullstellensatz
- Complete variety
- Elimination theory
- Quasiprojective variety
  - Gröbner basis
- Canonical bundle
- Complete intersection
- Serre duality
- Arithmetic genus, geometric genus, irregularity
Tangent space, Zariski tangent space
Function field
Ample vector bundle
Linear system of divisors
Birational geometry
- Blowing up
- Rational variety
- Unirational variety
Intersection number
- Intersection theory
- Serre's multiplicity conjectures
Albanese variety
Picard group
Pluricanonical ring
Modular form
Moduli space
Modular equation
- J-invariant
Algebraic function
Algebraic form
Addition theorem
Invariant theory
- Symbolic method of invariant theory
Geometric invariant theory
Toric geometry
Deformation theory
Singular point, non-singular
Singularity theory
- Newton polygon
Weil conjectures

Complex manifolds

Kähler manifold
Calabi-Yau manifold
Stein manifold
Hodge theory
Hodge cycle
Hodge conjecture
Algebraic geometry and analytic geometry
Mirror symmetry

Algebraic groups

Identity component
Linear algebraic group
- Additive group
- Multiplicative group
- Borel subgroup
- Parabolic subgroup
- Radical of an algebraic group
- Unipotent radical
- Lie-Kolchin theorem
- Mumford conjecture
Abelian variety
- Theta function
Grassmannian
Flag manifold
Algebraic torus
Weil restriction
Differential Galois theory

Contemporary foundations

Main article glossary of scheme theory

Commutative algebra

Prime ideal
Valuation (mathematics)
Regular local ring
Regular sequence (algebra)
Cohen-Macaulay ring
Gorenstein ring
Koszul complex
Spectrum of a ring
Zariski topology
Kähler differential

Sheaf theory

Locally ringed space
Coherent sheaf
Invertible sheaf
Sheaf cohomology
Hirzebruch-Riemann-Roch theorem
Grothendieck-Riemann-Roch theorem
Coherent duality

Schemes

Affine scheme
Scheme
Glossary of scheme theory
Éléments de géométrie algébrique
Grothendieck's Séminaire de géométrie algébrique
Flat morphism
Finite morphism
Quasi-finite morphism
Group scheme
Semistable elliptic curve
Grothendieck's relative point of view

الجيومطرقيون

Niels Henrik Abel
Carl Gustav Jakob Jacobi
Jakob Steiner
Julius Plücker
Bernhard Riemann
William Kingdon Clifford
Italian school of algebraic geometry
- Guido Castelnuovo
- Francesco Severi
Solomon Lefschetz
Oscar Zariski
Erich Kähler
W. V. D. Hodge
Kunihiko Kodaira
André Weil
Jean-Pierre Serre
Alexander Grothendieck
David Mumford
Igor Shafarevich
Heisuke Hironaka
Shigefumi Mori
Vladimir Voevodsky

الأسطح الجبرية

abelian surfaces (κ = 0) Two dimensional abelian varieties.
algebraic surfaces
Barlow surfaces General type, simply connected.
Barth sextic A degree-6 surface in P³ with 65 nodes.
Barth decic A degree-10 surface in P³ with 345 nodes.
Beauville surfaces General type
bielliptic surfaces (κ = 0) Same as hyperelliptic surfaces.
Bordiga surfaces A degree-6 embedding of the projective plane into P⁴ defined by the quartics throughعشرة points in general position.
Burniat surfaces General type
Campedelli surfaces General type
Castelnuovo surfaces General type
Catanese surfaces General type
class VII surfaces κ = −∞, non-algebraic.
Cayley surface Rational. A cubic surface with أربعة nodes.
Clebsch surface Rational. The surface Σx_i = Σx_i³ = 0 in P⁴.
cubic surfaces Rational.
Del Pezzo surfaces Rational. Anticanonical divisor is ample, for example P² blown up in at mostثمانية points.
Dolgachev surfaces Elliptic.
elliptic surfaces Surfaces with an elliptic fibration.
Enriques surfaces (κ = 0)
exceptional surfaces: Picard number has the maximal possible value h^1,1.
fake projective plane general type, found by Mumford, same betti numbers as projective plane.
Fano surfaces Rational. Same as del Pezzo surfaces.
Fermat surface of degree d: Solutions of w^d + x^d + y^d + z^d = 0 in P³.
general type κ = 2
Godeaux surfaces (general type)
Hilbert modular surfaces
Hirzebruch surfaces Rational ruled surfaces.
Hopf surfaces κ = −∞, non-algebraic, class VII
Horikawa surfaces general type
Horrocks-Mumford surfaces. These are certain abelian surfaces of degreeعشرة in P⁴, given as zero sets of sections of the rank 2 Horrocks-Mumford bundle.
Humbert surfaces These are certain surfaces in quotients of the Siegel upper half plane of genus 2.
hyperelliptic surfaces κ = 0, same as bielliptic surfaces.
Inoue surfaces κ = −∞, class VII,b₂ = 0. (Several quite different families were also found by Inoue, and are also sometimes called Inoue surfaces.)
Inoue-Hirzebruch surfaces κ = −∞, non-algebraic, type VII, b₂>0.
K3 surfaces κ = 0,
Kähler surfaces complex surfaces with a Kähler metric, which exists if and only if the first betti number b₁ is even.
Kodaira surfaces κ = 0, non-algebraic
Kummer surfaces κ = 0, special sorts of K3 surfaces.
minimal surfaces Surfaces with no rational −1 curves. (They have no connection with minimal surfaces in differential geometry.)
Mumford surface A "fake projective plane"
non-classical Enriques surface Only in characteristic 2.
numerical Campedelli surfaces surfaces of general type with the same Hodge numbers as a Campedelli surface.
numerical Godeaux surfaces surfaces of general type with the same Hodge numbers as a Godeaux surface.
projective plane Rational
properly elliptic surfaces κ = 1, elliptic surfaces of genus ≥2.
quadric surfaces Rational, isomorphic to P¹×P¹.
quartic surfaces Nonsingular ones are K3s.
quasi Enriques surface These only exist in characteristic 2.
quasi elliptic surface Only in characteristic p>0.
quotient surfaces: Quotients of surfaces by finite groups. Examples: Kummer, Godeaux, Hopf, Inoue surfaces.
rational surfaces κ = −∞, birational to projective plane
ruled surfaces κ = −∞
Sarti surface A degree-12 surface in P³ with 600 nodes.
Steiner surface A surface in P⁴ with singularities which is birational to the projective plane.
surface of general type κ = 2.
Togliatti surfaces Degree-5 surfaces in P³ with 31 nodes.
unirational surfaces Castelnuovo proved these are all rational in characteristic 0.
Veronese surface An embedding of the projective plane into P⁵.
Weddle surface κ = 0, birational to Kummer surface.
Zariski surfaces (only in characteristic p > 0): There is a purely inseparable dominant rational map of degree p from the projective plane to the surface.

انظر أيضاً

Enriques-Kodaira classification
Algebraic surface
قائمة الأسطح

Homology (mathematics)

Main article:Homology theory

Simplex
Simplicial complex
- Polytope
- Triangulation
- Barycentric subdivision
- Simplicial approximation theorem
- Abstract simplicial complex
- Simplicial set
- Simplicial category
Chain (mathematics)
Betti number
Euler characteristic
- Genus
- Riemann-Hurwitz formula
Singular homology
Cellular homology
Relative homology
Mayer-Vietoris sequence
Excision theorem
Universal coefficient theorem
Cohomology
- List of cohomology theories
- Cocycle class
- Cup product
- Cohomology ring
- De Rham cohomology
- Cech cohomology
- Alexander-Spanier cohomology
- Intersection cohomology
- Lyusternik-Schnirelmann category
Poincaré duality
Fundamental class
Applications
- Jordan curve theorem
- Brouwer fixed point theorem
- Invariance of domain
- Lefschetz fixed-point theorem
- Hairy ball theorem
- Degree (mathematics)
- Borsuk-Ulam Theorem
- Ham sandwich theorem
- Homology sphere

Homotopy theory

Homotopy
Path (topology)
Fundamental group
Homotopy group
Seifert-van Kampen theorem
Pointed space
Winding number
Simply connected
- Universal cover
Monodromy
Homotopy lifting property
Mapping cylinder
Mapping cone
Wedge sum
Smash product
Adjunction space
Cohomotopy
Cohomotopy group
Brown's representability theorem
Eilenberg-MacLane space
Fibre bundle
- Möbius strip
- Line bundle
  - Canonical line bundle
- Vector bundle
- Associated bundle
- Fibration
- Hopf bundle
- Classifying space
Cofibration
Homotopy groups of spheres
Plus construction
Whitehead theorem
Weak equivalence
Hurewicz theorem
H-space

اجازات اضافية

Künneth theorem
De Rham cohomology
Obstruction theory
Characteristic class
- Chern class
- Chern-Simons form
- Pontryagin class
- Pontryagin number
- Stiefel-Whitney class
Poincaré conjecture
Cohomology operation
- Steenrod algebra
Bott periodicity theorem
K-theory
- Topological K-theory
- Adams operation
- Algebraic K-theory
- Whitehead torsion
Cobordism
Thom space
Suspension functor
Stable homotopy theory
Spectrum (homotopy theory)
Morava K-theory
Hodge conjecture
Weil conjectures

Homological algebra

Chain complex
Commutative diagram
Exact sequence
- Five lemma
- Short five lemma
- Snake lemma
- Splitting lemma
- Extension problem
Spectral sequence
Abelian category
Group cohomology
Sheaf
- Sheaf cohomology
- Grothendieck topology
Derived category