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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. 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. The fundamental concepts of topology, such as continuity , compactness , and connectedness , can be defined in terms of open sets. Intuitively, continuous functions take nearby points to nearby points.

The most important of these invariants are homotopy groups , 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. Main article: Differential topology Differential topology is the field dealing with differentiable functions on 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 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—that is, one can smoothly "flatten out" certain manifolds, but it might require distorting the space and affecting the curvature or volume.

Main article: Geometric topology Geometric topology is a branch of topology that primarily focuses on low-dimensional manifolds i. In high-dimensional topology, characteristic classes are a basic invariant, and surgery theory is a key theory. Generalizations[ edit ] 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 of open sets as the basic notion of the theory, [19] 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.

These enzymes cut, twist, and reconnect the DNA, causing knotting with observable effects such as slower electrophoresis. 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. Computer science[ edit ] 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.

The main method used by topological data analysis is: Replace a set of data points with a family of simplicial complexes , indexed by a proximity parameter.

Analyse these topological complexes via algebraic topology — specifically, via the theory of persistent homology. The topological dependence of mechanical properties in solids is of interest in disciplines of mechanical engineering and materials science.

Electrical and mechanical properties depend on the arrangement and network structures of molecules and elementary units in materials. A topological quantum field theory or topological field theory or TQFT is a quantum field theory that computes topological invariants. 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.

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