nLab
topology

This page is about topology as a field of mathematics. For topology as a structure on a set, see topological space.

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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.

Related entries

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

• manifold, smooth manifold, generalized smooth space, diffeological space, smooth space, Froelicher space

• differential topology, microbundle, cobordism, derived smooth manifold

• low-dimensional topology

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 sets and related notions

• 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

• noncommutative topology

  • C-star-algebra, KK-theory

Computational Topology

• computational topology
• topological data analysis
• sources in computational topology