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I said earlier that what convergence was the starting point of topology. Why not take this seriously and replace topological spaces by spaces that would be defined in terms of notions of convergence directly , instead of through opens? This works well, both with Moore-Smith convergence in terms of nets and with filters.

One bonus is that we shall obtain Cartesian-closed categories, and this will be easy! Comparing with our exploration of exponential topologies Section 5. Oh, just to set it straight: there will be a snag with the net convergence approach. This will force us to use filters in the end. Let us implement our program of defining a replacement for topological spaces that would be based on convergence directly. Net convergence spaces form a category, whose morphisms are those maps that preserve convergence.

For those of you who are impatient, the announced snag lies somewhere in this very paragraph. Products in this category are defined in the obvious way: a net of tuples converges if and only if it converges componentwise. The problem is that nets on a set X do not form a set, but a proper class. But VBG set theory has no provision for forming a product of a proper class with anything: it is just too large. There are logical ways of fixing this. The latter was even used in topology.

All this is repaired by using filters. The class of filters on a set is a set , getting rid of the problem with nets. And of course, filters are enough to define convergence. We are no longer leaving the cosy realm of standard axioms for Mathematics. The only price we have to pay is to use filter convergence, which is more obscure than net convergence.

My reference is Hyland , who does wonders with them in the theory of computation. Variants of these limit spaces, convergence spaces already existed before, as one reckons by reading Hyland. These are the only axioms we are requiring. This is an example of a property that is true in every topological space, but may fail in filter spaces. Incidentally, filter spaces with this extra property are called convergence spaces.

One of the nice properties of filter spaces is that they form a Cartesian closed category. This is as for net convergence spaces… but this is really working now. This is componentwise convergence, as expected. This makes Flt a functor. This also defines a functor Top. Top and Flt are not inverses of each other: in fact there are many more filter spaces than topological spaces.

But Top is left adjoint to Flt , so that one can say that Top X is the free topological space over the filter space X. At the level of points, it is just again the identity map, but more filters converge in Flt Top X than in X. This allows us to consider every topological space X as a topological filter space, equating X with Top X. But the circle is not homeomorphic to the doughnut. While topological spaces can be extremely varied and exotic, many areas of topology focus on the more familiar class of spaces known as manifolds.

A manifold is a topological space that resembles Euclidean space near each point. More precisely, each point of an n -dimensional manifold has a neighborhood that is homeomorphic to the Euclidean space of dimension n. Lines and circles , but not figure eights , are one-dimensional manifolds.

Two-dimensional manifolds are also called surfaces , although not all surfaces are manifolds. 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. 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. 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.

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Changing a topology consists of changing the collection of open sets, and this changes which functions are continuous, and which subsets are compact or connected. Metric spaces are an important class of topological spaces where distances between any two points are 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 r centered at x is the set of the points whose distance to x is less than d.

Many common spaces are topological space whose topology can be defined by a metric. This is the case of the real line , the complex plane , real and complex vector spaces and Euclidean spaces. Having a metric simplifies many proofs.

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5. Algebraic topology is a branch of mathematics that uses tools from algebra to study topological spaces. 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. 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.

## Quantum Algebra and Quantum Topology Seminar | Problems in Mathematics

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. 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,  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. Knot theory , a branch of topology, is used in biology to study the effects of certain enzymes on DNA. 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.

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4. 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:. Topology is relevant to physics in areas such as condensed matter physics ,  quantum field theory and physical cosmology. 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.

## Questions in 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. Donaldson , Jones , Witten , and Kontsevich have all won Fields Medals for work related to topological field theory.

The topological classification of Calabi-Yau manifolds has important implications in string theory , as different manifolds can sustain different kinds of strings. In cosmology, topology can be used to describe the overall shape of the universe. The possible positions of a robot can be described by a manifold called configuration space. These paths represent a motion of the robot's joints and other parts into the desired pose. Tanglement puzzles are based on topological aspects of the puzzle's shapes and components. In order to create a continuous join of pieces in a modular construction, it is necessary to create an unbroken path in an order which surrounds each piece and traverses each edge only once.

This process is an application of the Eulerian path.