# topology

•  Concepts Topologies on sets Main article: Topological space The term topology also refers to a specific mathematical idea central to the area of mathematics called
topology.

• 2-dimensional topology can be studied as complex geometry in one variable (Riemann surfaces are complex curves) – by the uniformization theorem every conformal class of 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.

• A topological space is a set endowed with a structure, called a topology, which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity.

• Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible.

• Computer science Topological data analysis uses techniques from algebraic topology to determine the large scale structure of a set (for instance, determining if a cloud
of points is spherical or toroidal).

• In pointless topology one considers instead the lattice of open sets as the basic notion of the theory, while Grothendieck topologies are structures defined on arbitrary
categories that allow the definition of sheaves on those categories, and with that the definition of general cohomology theories.

• Main articles: Continuous function and homeomorphism A function or map from one topological space to another is called continuous if the inverse image of any open set is open.

• In addition to establishing the basic ideas of set theory, Cantor considered point sets in Euclidean space as part of his study of Fourier series.

• Metric spaces are an important class of topological spaces where the distance between any two points is defined by a function called a metric.

• In 1895, he published his ground-breaking paper on Analysis Situs, which introduced the concepts now known as homotopy and homology, which are now considered part of algebraic
topology.

• Although TQFTs were invented by physicists, they are also of mathematical interest, being related to, among other things, knot theory, the theory of four-manifolds in algebraic
topology, and to the theory of moduli spaces in algebraic geometry.

• If the function maps the real numbers to the real numbers (both spaces with the standard topology), then this definition of continuous is equivalent to the definition of continuous
in calculus.

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

•  A metric space is now considered a special case of a general topological space, with any given topological space potentially giving rise to many distinct metric spaces.

• The basic object of study is topological spaces, which are sets equipped with a topology, that is, a family of subsets, called open sets, which is closed under finite intersections
and (finite or infinite) unions.

• Several topologies can be defined on a given space.

• The following are basic examples of topological properties: the dimension, which allows distinguishing between a line and a surface; compactness, which allows distinguishing
between a line and a circle; connectedness, which allows distinguishing a circle from two non-intersecting circles.

• Manifolds Main article: Manifold While topological spaces can be extremely varied and exotic, many areas of topology focus on the more familiar class of spaces known
as manifolds.

• More specifically, differential topology considers the properties and structures that require only a smooth structure on a manifold to be defined.

• In a metric space, an open set is a union of open disks, where an open disk of radius r centered at x is the set of all points whose distance to x is less than r. Many common
spaces are topological spaces whose topology can be defined by a metric.

• Examples include the plane, the sphere, and the torus, which can all be realized without self-intersection in three dimensions, and the Klein bottle and real projective plane,
which cannot (that is, all their realizations are surfaces that are not manifolds).

• If a continuous function is one-to-one and onto, and if the inverse of the function is also continuous, then the function is called a homeomorphism and the domain of the function
is said to be homeomorphic to the range.

• Another way of saying this is that the function has a natural extension to the topology.

• For instance, volume and Riemannian curvature are invariants that can distinguish different geometric structures on the same smooth manifold – that is, one can smoothly “flatten
out” certain manifolds, but it might require distorting the space and affecting the curvature or volume.

•  It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.

• For instance, the real line, the complex plane, and the Cantor set can be thought of as the same set with different topologies.

• This is the case of the real line, the complex plane, real and complex vector spaces and Euclidean spaces.

• The main method used by topological data analysis is to: 1.

• His 1736 paper on the Seven Bridges of Königsberg is regarded as one of the first practical applications of topology.

• Geometric topology Main article: Geometric topology Geometric topology is a branch of topology that primarily focuses on low-dimensional manifolds (that is, spaces of
dimensions 2, 3, and 4) and their interaction with geometry, but it also includes some higher-dimensional topology.

• This fact is immediately convincing to most people, even though they might not recognize the more formal statement of the theorem, that there is no nonvanishing continuous
tangent vector field on the sphere.

• In neuroscience, topological quantities like the Euler characteristic and Betti number have been used to measure the complexity of patterns of activity in neural networks.

•  Topology is of further significance in Contact mechanics where the dependence of stiffness and friction on the dimensionality of surface structures is the subject of
interest with applications in multi-body physics.

• Motivation The motivating insight behind topology is that some geometric problems depend not on the exact shape of the objects involved, but rather on the way they are put
together.

•  In cosmology, topology can be used to describe the overall shape of the universe.

• In condensed matter a relevant application to topological physics comes from the possibility to obtain one-way current, which is a current protected from backscattering.

• For example, the square and the circle have many properties in common: they are both one dimensional objects (from a topological point of view) and both separate the plane
into two parts, the part inside and the part outside.

• 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.

• Topics General topology Main article: General topology General topology is the branch of topology dealing with the basic set-theoretic definitions and constructions
used in topology.

• Another name for general topology is point-set topology.

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Photo credit: https://www.flickr.com/photos/macieklew/537625768/’] 