nLab
topology
This page is about topology as a field of mathematics . For topology as a structure on a set , see topological space .
Contents
Idea
Topology is one of the basic fields of mathematics . The term is also used for a particular structure in a topological space ; see topological structure for that.
The subject of topology deals with the expressions of continuity and boundary , and studying the geometric properties of (originally, metric) spaces and relations of subspaces, which do not change under continuous deformations, regardless to the changes in their metric properties. Topology as a structure enables one to model continuity and convergence locally. More recently, in metric spaces, topologists and geometric group theorists started looking at asymptotic properties at large, which are in some sense dual to the standard topological structure and are usually referred to as coarse topology .
There are many cousins of topological spaces, e.g. sites, locales, topoi, higher topoi, uniformity spaces and so on which specialize or generalize some aspect or structure usually found in Top . One of the tools of topology, homotopy theory , has long since crossed the boundaries of topology and applies to many other areas, thanks to many examples and motivations as well as of abstract categorical frameworks for homotopy like Quillen model categories , Brown’s categories of fibrant objects and so on.
Here we exclude the entries on topos theory which may also be viewed as an approach to topology (but also geometry, logic…) as it is comprehensively linked from entries topos , topos theory , infinity topos , Grothendieck topology , site .
Topological spaces
Top CW-complex , general topology
induced topology , subspace , interior , boundary , closure
sphere , metric space , metrizable space
topological space , continuous map , homeomorphism , neighborhood
pointed space , contractible space , connected space , second countable space
convergence space , pretopological space , pseudotopological space , coarse topology
metric space , filtered space , connected filtered space , complete space , net , Polish space
separation axioms , Hausdorff space , regular space , normal space
noetherian topological space , irreducible topological space
compact space , locally compact space , compactum , paracompact space
Frechet-Uryson space , sequential space , uniform space
convenient category of topological spaces , compactly generated space
nice topological space , nice category of spaces
pointless topology , locale , cover , site , ringed space sphere
Sierpinski space , Warsaw circle
See also examples in topology .
Manifolds and generalizations
Algebraic topology and homotopy theory in general
homotopy , homotopy inverse , homotopy theory , homotopy equivalence , rational homotopy theory
shape theory , algebraic topology , basic problems of algebraic topology
homotopy limit , homotopy colimit , homotopy pullback , homotopy image , homotopy coherent category theory , homotopy coherent nerve , Brown representability theorem
homotopy type , homotopy 1-type , homotopy 2-type , homotopy 3-type , homotopy n-type
deformation retract , neighborhood retract , Postnikov system
fundamental group , homotopy group , suspension , Freudenthal suspension theorem , delooping
loop space , free loop space object ,
cup product , Dold-Thom theorem
topological group , H-space , co-H-space
principal bundle , fiber bundle , vector bundle , principal 2-bundle , trivial bundle
fibration , Hurewicz fibration , Hurewicz connection , homotopy lifting property , Dold fibration
homotopy extension property , Hurewicz cofibration , deformation retract , model structure on topological spaces , Strøm's theorem
cocylinder , cylinder object , path object , mapping cone , path groupoid , path n-groupoid , fundamental groupoid
classifying space , Eilenberg-MacLane space ,Moore space , Moore path category
spectrum , smash product of spectra , symmetric spectrum , stable (infinity,1)-category , commutative ring spectrum
model category , model 2-category , model stack , Quillen adjunction , Quillen equivalence , category with weak equivalences , category of fibrant objects
Reedy category , Reedy model structure , generalized Reedy category
model structure on chain complexes , model structure on crossed complexes
homotopy hypothesis , homotopy theory of Grothendieck , test category
simplicial set , simplex category , simplicial identities , simplicial object , cosimplicial object
simplicial complex , singular simplicial complex ,
boundary of a simplex , simplicial skeleton , category of simplices
combinatorial spectrum , simplicial groupoid , simplicial model category
simplicially enriched category , quasicategory , Segal category , complete Segal space
Kan fibration , Kan complex , nerve , nerve and realization , simplicial homotopy , simplicial homotopy group , simplicial local system
simplicial presheaf , local model structure on simplicial presheaves , local model structure on simplicial sheaves
marked simplicial set , model structure on marked simplicial over-sets , infinity topos , Higher Topos Theory
dendroidal set , model structure on dendroidal sets , cubical set , cellular set , cyclic object , Theta space
Sheaves, stacks, cohomology
sheaf , flabby sheaf , local epimorphism , etale space , hypercover
cohomology , sheaf cohomology , abelian sheaf cohomology , local system , category of sheaves
Čech cohomology , Bredon cohomology , twisted cohomology , twisting cochain
nonabelian algebraic topology , nonabelian cocycle , nonabelian cohomology
principal infinity-bundle , BrownAHT , category of fibrant objects
topological K-theory , gerbe , twisted K-theory , Karoubi K-theory , differential K-theory , K-theory spectrum
de Rham cohomology , string topology
Thom space , Thom bundle? , fiber integration , equivariant cohomology
Poincaré duality , Spanier-Whitehead duality
generalized cohomology , generalized (Eilenberg-Steenrod) cohomology , generalized (Eilenberg-Steenrod) homotopy
topological stack , orbispace
topological quantum field theory , tmf , cobordism hypothesis , topological T-duality
characteristic class , Chern-Weil theory
Non-commutative topology
Computational Topology
References
Topospaces , a Wiki with basic material on topology.
Revised on May 11, 2016 08:19:02
by
Urs Schreiber
(131.220.184.222)