Instruct: Given a question, retrieve documents that can help answer the question.
Query: What is Topology?
In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, twisting, crumpling and bending, but not tearing or gluing.
The term "Grothendieck topology" has changed in meaning. In it meant what is now called a Grothendieck pretopology, and some authors still use this old meaning. modified the definition to use sieves rather than covers. Much of the time this does not make much difference, as each Grothendieck pretopology determines a unique Grothendieck topology, though quite different pretopologies can give the same topology.
The topology of an electronic circuit is the form taken by the network of interconnections of the circuit components. Different specific values or ratings of the components are regarded as being the same topology. Topology is not concerned with the physical layout of components in a circuit, nor with their positions on a circuit diagram. It is only concerned with what connections exist between the components. There may be numerous physical layouts and circuit diagrams that all amount to the same topology.
The fundamental concepts in point-set topology are continuity, compactness, and connectedness. 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. If we change the definition of open set, we change what continuous functions, compact sets, and connected sets are. Each choice of definition for open set is called a topology. A set with a topology is called a topological space.
The separation axioms are axioms only in the sense that, when defining the notion of topological space, one could add these conditions as extra axioms to get a more restricted notion of what a topological space is. The modern approach is to fix once and for all the axiomatization of topological space and then speak of "kinds" of topological spaces.
However, the term "separation axiom" has stuck. The separation axioms are denoted with the letter "T" after the German "Trennungsaxiom", which means "separation axiom."
An interesting question is to ask, given a symmetry class and a dimension of the Brillouin zone, what are all the equivalence classes of Hamiltonians. Each equivalence class can be labeled by a topological invariant; two Hamiltonians whose topological invariant are different cannot be deformed into each other and belong to different equivalence classes.
"Natural topology" does quite often have a more specific meaning, at least given some prior contextual information: the natural topology is a topology which makes a natural map or collection of maps continuous. This is still imprecise, even once one has specified what the natural maps are, because there may be many topologies with the required property. However, there is often a finest or coarsest topology which makes the given maps continuous, in which case these are obvious candidates for "the" natural topology.
A GIS can recognize and analyze the spatial relationships that exist within digitally stored spatial data. These topological relationships allow complex spatial modelling and analysis to be performed. Topological relationships between geometric entities traditionally include adjacency (what adjoins what), containment (what encloses what), and proximity (how close something is to something else).
Topological indistinguishability of points is an equivalence relation. No matter what topological space "X" might be to begin with, the quotient space under this equivalence relation is always T. This quotient space is called the Kolmogorov quotient of "X", which we will denote KQ("X"). Of course, if "X" was T to begin with, then KQ("X") and "X" are naturally homeomorphic.
Categorically, Kolmogorov spaces are a reflective subcategory of topological spaces, and the Kolmogorov quotient is the reflector.
Unifying the work on function spaces of Georg Cantor, Vito Volterra, Cesare Arzelà, Jacques Hadamard, Giulio Ascoli and others, Maurice Fréchet introduced the metric space in 1906.[9] 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. In 1914, Felix Hausdorff coined the term "topological space" and gave the definition for what is now called a Hausdorff space.[10] Currently, a topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski.[11]
At the beginning of the 20th century, Henri Poincaré was working on the foundations of topology—what would later be called combinatorial topology and then algebraic topology. He was particularly interested in what topological properties characterized a sphere.
In the mid-seventies, he extended the connection between geometric topology and algebraic K-theory by introducing a kind of algebraic K-theory for topological spaces. This led to new foundations for algebraic K-theory (using what are now called Waldhausen categories) and also gave new impetus to the study of highly structured ring spectra. Articles: "Algebraic K-Theory of Topological Spaces I" (1976) and "Algebraic K-theory of spaces" (1983).
The space of distributions on "U" is denoted by D′("U") and it is the continuous dual space of D("U"). No matter what dual topology is placed on D′("U"), a "sequence" of distributions converges in this topology if and only if it converges pointwise (although this need not be true of a net), which is why the topology is sometimes defined to be the weak-* topology. But often the topology of bounded convergence, which in this case is the same as the topology of uniform convergence on compact sets, is placed on D′("U") since it is with this topology that D′("U") becomes a nuclear Montel space and it is with this topology that the kernels theorem of Schwartz holds. No matter which topology is chosen, D′("U") will be a non-metrizable, locally convex topological vector space.
The construct Top is the model of what is called a topological category. These categories are characterized by the fact that every structured source formula_1 has a unique initial lift formula_2. In Top the initial lift is obtained by placing the initial topology on the source. Topological categories have many properties in common with Top (such as fiber-completeness, discrete and indiscrete functors, and unique lifting of limits).
A geographic information system (GIS) can recognize and analyze the spatial relationships that exist within digitally stored spatial data. These topological relationships allow complex spatial modelling and analysis to be performed. Topological relationships between geometric entities traditionally include adjacency (what adjoins what), containment (what encloses what), and proximity (how close something is to something else).
In what follows "X" denotes a locally compact topological space. The continuous real-valued functions with compact support on "X" form a vector space formula_1, which can be given a natural locally convex topology. Indeed, formula_2 is the union of the spaces formula_3 of continuous functions with support contained in compact sets "K". Each of the spaces formula_3 carries naturally the topology of uniform convergence, which makes it into a Banach space. But as a union of topological spaces is a special case of a direct limit of topological spaces, the space formula_2 can be equipped with the direct limit locally convex topology induced by the spaces formula_3; this topology is finer than the topology of uniform convergence.
There are several other topologies on the space of bounded operators, and one can ask what are the *-algebras closed in these topologies. If is closed in the norm topology then it is a C*-algebra, but not necessarily a von Neumann algebra. One such example is the C*-algebra of compact operators (on an infinite dimensional Hilbert space). For most other common topologies the closed *-algebras containing 1 are still von Neumann algebras; this applies in particular to the weak operator, strong operator, *-strong operator, ultraweak, ultrastrong, and *-ultrastrong topologies.
It is related to the Jacobson density theorem.
There is a standard construction for introducing a topology on the continuous dual formula_2 of a topological vector space formula_19. Fix a collection formula_46 of bounded subsets of formula_19. Then one has the topology on formula_19 of uniform convergence on sets from formula_46, or what is the same thing, the topology generated by seminorms of the form 
where formula_51 is a continuous linear functional on formula_19, and formula_53 runs over the class formula_46.
Equivariant topology List of algebraic topology topics List of examples in general topology List of general topology topics List of geometric topology topics List of topology topics Publications in topology Topoisomer Topology glossary Topological geometry Topological order
Convergence is the state of a set of routers that have the same topological information about the internetwork in which they operate. For a set of routers to have "converged", they must have collected all available topology information from each other via the implemented routing protocol, the information they gathered must not contradict any other router's topology information in the set, and it must reflect the real state of the network. In other words: In a converged network all routers "agree" on what the network topology looks like.
Let "X" be a vector space and let "Y" be a vector subspace of the algebraic dual of "X" that separates points on "X". Any locally convex Hausdorff topological vector space (TVS) topology on "X" with the property that when "X" is equipped with this topology has "Y" as its continuous dual space is said to be compatible with duality between "X" and "Y". If we give "X" the weak topology formula_304 then formula_305 is a Hausdorff locally convex topological vector space (TVS) and formula_304 is compatible with duality between "X" and "Y" (i.e. formula_307). We can now ask the question: what are "all" of the locally convex Hausdorff TVS topologies that we can place on "X" that are compatible with duality between "X" and "Y"? The answer to this question is called the Mackey–Arens theorem:
Theorem. Let "X" be a vector space and let formula_308 be a locally convex Hausdorff topological vector space topology on "X". Let formula_90 denote the continuous dual space of "X" and let formula_310 denote "X" with the topology formula_308. Then the following are equivalent:
 is identical to formula_90.
And furthermore, 
^*, and
A quasi-coherent sheaf is roughly one that looks locally like the sheaf of a module over a ring. The first problem is to decide what one means by "locally": this involves the choice of a Grothendieck topology, and there are many possible choices for this, all of which have some problems and none of which seem completely satisfactory. The Grothendieck topology should be strong enough so that the stack is locally affine in this topology: schemes are locally affine in the Zariski topology so this is a good choice for schemes as Serre discovered, algebraic spaces and Deligne–Mumford stacks are locally affine in the etale topology so one usually uses the etale topology for these, while algebraic stacks are locally affine in the smooth topology so one can use the smooth topology in this case. For general algebraic stacks the etale topology does not have enough open sets: for example, if G is a smooth connected group then the only etale covers of the classifying stack BG are unions of copies of BG, which are not enough to give the right theory of quasicoherent sheaves.
In general, the product of the topologies of each "X" forms a basis for what is called the box topology on "X". In general, the box topology is finer than the product topology, but for finite products they coincide.
In general, the product of the topologies of each "X" forms a basis for what is called the box topology on "X". In general, the box topology is finer than the product topology, but for finite products they coincide.
A non-trivial topology does not in itself imply energetic stability. There is in fact no necessary relation between topology and energetic stability. Hence, one must be careful not to confuse ‘topological stability,’ which is a mathematical concept, with energy stability in real physical systems. Topological stability refers to the idea that in order for a system described by a continuous field to transition from one topological state to another, a rupture must occur in the continuous field, i.e. a discontinuity must be produced. For example, if one wishes to transform a flexible balloon doughnut (torus) into an ordinary spherical balloon, it is necessary to introduce a rupture on some part of the balloon doughnut's surface. Mathematically, the balloon doughnut would be described as 'topologically stable.' However, in physics, the free energy required to introduce a rupture enabling the transition of a system from one ‘topological’ state to another is always "finite". For example, it is possible to turn a rubber ballon into flat piece of rubber by poking it with a needle (and popping it!). Thus, while a physical system can be "approximately" described using the mathematical concept of topology, attributes such as "energetic" stability are dependent on the system's parameters—the strength of the rubber in the example above—not the topology per se. In order to draw a meaningful parallel between the concept of topological stability and the energy stability of a system, the analogy must necessarily be accompanied by the introduction of a non-zero phenomenological ‘field rigidity’ to account for the finite energy needed to rupture the field’s topology. Modeling and then integrating this field rigidity can be likened to calculating a breakdown energy-density of the field. These considerations suggest that what is often referred to as ‘topological protection,’ or a 'topological barrier,' should more accurately be referred to as a 'topology-related energy barrier,' though this terminology is somewhat cumbersome. A quantitative evaluation of such a topological barrier can be obtained by extracting the critical magnetic configuration when the topological number changes during the dynamical process of a skyrmion creation event. Applying the topological charge defined in a lattice, the barrier height is theoretically shown to be proportional to the exchange stiffness.