قائمة مواضيع الطوبولوجيا
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Basic concepts
- فضاء طوبولوجي Topological space
- خاصية طوبولوجية Topological property
- مجموعة مفتوحة
- مجموعة مغلقة closed set
- Continuity (topology)
- هوميومورفية Homeomorphism
- هوميومورفية محلية Local homeomorphism
- Open and closed maps
- Germ (mathematics)
- Base (topology), subbase
- Open cover
- Atlas (topology)
Limits
Topological properties
Compactness and countability
- Compact space
- 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
Separation axioms
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
- Trivial topology
- Cofinite topology
- Finer topology
- Product topology
- 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
Metric spaces
Topology and order theory
Descriptive set theory
Dimension theory
See also main article dimension
Topological algebra
Combinatorial topology
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
Differential geometry of surfaces
Calculus on manifolds
See also multivariable calculus, list of multivariable calculus topics
Differential topology
Fiber bundles
Riemannian geometry
- List of coordinate charts
- Metric tensor
- Riemannian manifold
- Non-Euclidean geometry
- Hodge star operator
- Killing vector field
- 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
- 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
Other
Classical topics in projective geometry
- Affine space
- Projective space
- Projective line, cross-ratio
- 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
- Hyperelliptic curve
- Klein quartic
- modular curve
- 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
- Hurwitz's automorphisms theorem
- Clifford's theorem
- Gonality of an algebraic curve
- Weil's reciprocity law
- Goppa code
Algebraic surfaces
Algebraic geometry: classical approach
- Algebraic variety
- Tangent space, Zariski tangent space
- Function field
- Ample vector bundle
- Linear system of divisors
- Birational geometry
- Intersection number
- Albanese variety
- Picard group
- Pluricanonical ring
- Modular form
- Moduli space
- Modular equation
- Algebraic function
- Algebraic form
- Addition theorem
- Invariant theory
- Geometric invariant theory
- Toric geometry
- Deformation theory
- Singular point, non-singular
- Singularity theory
- Weil conjectures
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Complex manifolds
Algebraic groups
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
Category theory
الجيومطرقيون
- Niels Henrik Abel
- Carl Gustav Jakob Jacobi
- Jakob Steiner
- Julius Plücker
- Bernhard Riemann
- William Kingdon Clifford
- Italian school of algebraic geometry
- 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 through 10 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 4 nodes.
- Clebsch surface Rational. The surface Σxi = Σxi3 = 0 in P4.
- cubic surfaces Rational.
- Del Pezzo surfaces Rational. Anticanonical divisor is ample, for example P2 blown up in at most 8 points.
- Dolgachev surfaces Elliptic.
- elliptic surfaces Surfaces with an elliptic fibration.
- Enriques surfaces (κ = 0)
- exceptional surfaces: Picard number has the maximal possible value h1,1.
- 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 wd + xd + yd + zd = 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 degree 10 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
Homology (mathematics)
Main article:Homology theory
Homotopy theory
- Homotopy
- Path (topology)
- Fundamental group
- Homotopy group
- Seifert-van Kampen theorem
- Pointed space
- Winding number
- Simply connected
- 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
- Cofibration
- Homotopy groups of spheres
- Plus construction
- Whitehead theorem
- Weak equivalence
- Hurewicz theorem
- H-space