Editing Topology (section)

==Topics==

===General topology===
{{Main|General topology}}
General topology is the branch of topology dealing with the basic set-theoretic definitions and constructions used in topology.<ref>Munkres, James R. Topology. Vol. 2. Upper Saddle River: Prentice Hall, 2000.</ref><ref>Adams, Colin Conrad, and Robert David Franzosa. Introduction to topology: pure and applied. Pearson Prentice Hall, 2008.</ref> It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology.

The basic object of study is [[topological space]]s, which are sets equipped with a [[topology (structure)|topology]], that is, a family of [[subset]]s, called ''open sets'', which is [[closure (mathematics)|closed]] under finite [[set intersection|intersection]]s and (finite or infinite) [[set union|union]]s. The fundamental concepts of topology, such as ''[[continuity (mathematics)|continuity]]'', ''[[compactness]]'', and ''[[connectedness]]'', can be defined in terms of open sets. Intuitively, continuous functions take nearby points to nearby points. Compact sets are those that can be covered by finitely many sets of arbitrarily small size. Connected sets are sets that cannot be divided into two pieces that are far apart. The words ''nearby'', ''arbitrarily small'', and ''far apart'' can all be made precise by using open sets. Several topologies can be defined on a given space. Changing a topology consists of changing the collection of open sets. This changes which functions are continuous and which subsets are compact or connected.

[[Metric space]]s are an important class of topological spaces where the distance between any two points is defined by a function called a ''metric''. In a metric space, an open set is a union of open disks, where an open disk of radius {{mvar|r}} centered at {{mvar|x}} is the set of all points whose distance to {{mvar|x}} is less than {{mvar|r}}. Many common spaces are topological spaces whose topology can be defined by a metric. This is the case of the [[real line]], the [[complex plane]], real and complex [[vector space]]s and [[Euclidean space]]s. Having a metric simplifies many proofs.

===Algebraic topology===
{{Main|Algebraic topology}}
Algebraic topology is a branch of mathematics that uses tools from [[algebra]] to study topological spaces.<ref>Allen Hatcher, [http://pi.math.cornell.edu/~hatcher/AT/ATpage.html ''Algebraic topology.''] {{Webarchive|url=https://web.archive.org/web/20120206155217/http://www.math.cornell.edu/~hatcher/AT/ATpage.html |date=6 February 2012 }} (2002) Cambridge University Press, xii+544 pp.&nbsp;{{isbn|0-521-79160-X|0-521-79540-0}}.</ref> The basic goal is to find algebraic invariants that [[classification theorem|classify]] topological spaces [[up to]] homeomorphism, though usually most classify up to homotopy equivalence.

The most important of these invariants are [[homotopy group]]s, homology, and [[cohomology]].

Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a [[free group]] is again a free group.

===Differential topology===
{{Main|Differential topology}}
Differential topology is the field dealing with [[differentiable function]]s on [[differentiable manifold]]s.<ref>{{cite book | first = John M. | last = Lee | year = 2006 | title = Introduction to Smooth Manifolds | publisher = Springer-Verlag | isbn = 978-0-387-95448-6}}</ref> It is closely related to [[differential geometry]] and together they make up the geometric theory of differentiable manifolds.

More specifically, differential topology considers the properties and structures that require only a [[smooth structure]] on a manifold to be defined. Smooth manifolds are "softer" than manifolds with extra geometric structures, which can act as obstructions to certain types of equivalences and [[Deformation theory|deformations]] that exist in differential topology. For instance, volume and [[Riemannian curvature]] are invariants that can distinguish different geometric structures on the same smooth manifold{{snd}}that is, one can smoothly "flatten out" certain manifolds, but it might require distorting the space and affecting the curvature or volume.

===Geometric topology===
{{Main|Geometric topology}}

Geometric topology is a branch of topology that primarily focuses on low-dimensional [[manifold]]s (that is, spaces of dimensions 2, 3, and 4) and their interaction with geometry, but it also includes some higher-dimensional topology.<ref>R. B. Sher and [[R. J. Daverman]] (2002), ''Handbook of Geometric Topology'', North-Holland. {{isbn|0-444-82432-4}}</ref> Some examples of topics in geometric topology are [[Orientable manifold|orientability]], [[handle decomposition]]s, [[local flatness]], crumpling and the planar and higher-dimensional [[Jordan-Schönflies theorem|Schönflies theorem]].

In high-dimensional topology, [[characteristic classes]] are a basic invariant, and [[surgery theory]] is a key theory.

Low-dimensional topology is strongly geometric, as reflected in the [[uniformization theorem]] in 2 dimensions – every surface admits a constant curvature metric; geometrically, it has one of 3 possible geometries: positive [[curvature]]/spherical, zero curvature/flat, and negative curvature/hyperbolic – and the [[geometrization conjecture]] (now theorem) in 3 dimensions – every 3-manifold can be cut into pieces, each of which has one of eight possible geometries.

2-dimensional topology can be studied as [[complex geometry]] in one variable ([[Bernhard Riemann|Riemann]] surfaces are complex curves) – by the uniformization theorem every [[Conformal geometry|conformal class]] of [[Metric (mathematics)|metrics]] is equivalent to a unique complex one, and 4-dimensional topology can be studied from the point of view of complex geometry in two variables (complex surfaces), though not every 4-manifold admits a complex structure.

===Generalizations===
Occasionally, one needs to use the tools of topology but a "set of points" is not available. In [[pointless topology]] one considers instead the [[lattice (order)|lattice]] of open sets as the basic notion of the theory,<ref>{{cite journal | last1 = Johnstone | first1 = Peter T. | author-link = Peter Johnstone (mathematician) | year = 1983 | title = The point of pointless topology | url = http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.bams/1183550014 | journal = Bulletin of the American Mathematical Society | volume = 8 | issue = 1 | pages = 41–53 | doi = 10.1090/s0273-0979-1983-15080-2 | doi-broken-date = 30 April 2025 | doi-access = free | archive-date = 27 February 2021 | access-date = 13 January 2014 | archive-url = https://web.archive.org/web/20210227011239/https://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.bams%2F1183550014 | url-status = live }}</ref> while [[Grothendieck topology|Grothendieck topologies]] are structures defined on arbitrary [[category theory|categories]] that allow the definition of [[sheaf (mathematics)|sheaves]] on those categories and with that the definition of general cohomology theories.<ref>{{Cite book | last1=Artin | first1=Michael | author1-link=Michael Artin | title=Grothendieck topologies | publisher=Harvard University, Dept. of Mathematics | year=1962 | zbl=0208.48701 | location=Cambridge, MA }}</ref>