( see figure ) The effect was observed to be independent of parameters such as the system size and impurities, and in 1981, theorist Robert Laughlin proposed a theory describing the integer states in terms of a topological invariant called the Chern number.

In topology, the Tietze extension theorem states that, if X is a normal topological space and

In topology, Urysohn's lemma is a lemma that states that a topological space is normal if and only if any two disjoint closed subsets can be separated by a function.

Urysohn's lemma states that a topological space is normal if and only if any two disjoint closed sets can be separated by a continuous function.

In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact.

For example, the quantum finite automaton or topological automaton has uncountable infinity of states.

* Open mapping theorem ( topological groups ) states that a surjective continuous homomorphism of a locally compact Hausdorff group G onto a locally compact Hausdorff group H is an open mapping if G is σ-compact.

However, this assumption does leave out finite energy states like solitons which can't be generated by a polynomial of fields smeared by test functions because a soliton, at least from a field theoretic perspective, is a global structure involving topological boundary conditions at infinity.

In differential geometry, the Atiyah – Singer index theorem, proved by, states that for an elliptic differential operator on a compact manifold, the analytical index ( related to the dimension of the space of solutions ) is equal to the topological index ( defined in terms of some topological data ).

The free product is important in algebraic topology because of van Kampen's theorem, which states that the fundamental group of the union of two path-connected topological spaces is always an amalgamated free product of the fundamental groups of the spaces.

In condensed matter physics, topological quantum field theories are the low energy effective theories of topologically ordered states, such as fractional quantum Hall states, string-net condensed states, and

The topological properties of the system have a major impact on the properties of the density of states.

He is best known for his work on topology, including the metrization theorem he proved in 1926, and the Tychonoff's theorem, which states that every product of arbitrarily many compact topological spaces is again compact.

It states that an analytic subspace of complex projective space that is closed ( in the ordinary topological sense ) is an algebraic subvariety.

The Tikhonov ( Tychonoff ) fixed point theorem is applied to any locally convex topological vector space V. It states that for any non-empty compact convex set X in V, any continuous function

In mathematics, the Hilbert – Smith conjecture is concerned with the transformation groups of manifolds ; and in particular with the limitations on topological groups G that can act effectively ( faithfully ) on a ( topological ) manifold M. Restricting to G which are locally compact and have a continuous, faithful group action on M, it states that G must be a Lie group.

In homotopy theory ( a branch of mathematics ), the Whitehead theorem states that if a continuous mapping f between topological spaces X and Y induces isomorphisms on all homotopy groups, then f is a homotopy equivalence provided X and Y are connected and have the homotopy-type of CW complexes.

The theorem states that a topological space is metrizable if and only if it is regular and Hausdorff and has a countably locally finite ( i. e., σ-locally finite ) basis.

Although the LSZ reduction formula cannot handle bound states, massless particles and topological solitons, it can be generalized to cover bound states, by use of composite fields which are often nonlocal.

However, the definition of " Hilbert space " can be broadened to accommodate these states ( see the Gelfand – Naimark – Segal construction or rigged Hilbert spaces ).

* Wave functions and other quantum states can be represented as vectors in a complex Hilbert space.

( Technically, the quantum states are rays of vectors in the Hilbert space, as corresponds to the same state for any nonzero complex number c .)

* Measurements are associated with linear operators ( called observables ) on the Hilbert space of quantum states.

Different configurations or states of the body correspond to different regions in Euclidean space.

Moreover, the Banach fixed point theorem states that every contraction mapping on a nonempty complete metric space has a unique fixed point, and that for any x in M the iterated function sequence x, f ( x ), f ( f ( x )), f ( f ( f ( x ))), ... converges to the fixed point.

The Big Bang theory states that it is the point in which all dimensions came into existence, the start of both space and time.

Additionally, the costs of superpower status — the military, space program, subsidies to client states — were out of proportion to the Soviet economy.

The Banach fixed point theorem states that a contraction mapping on a complete metric space admits a fixed point.

In 1926 von Neuman showed that if atomic states were understood as vectors in Hilbert space, then they would correspond with both Schrödinger's wave function theory and Heisenberg's matrices.

The states with the same energy form an energy shell Ω, a sub-manifold of the phase space.

Decoherent interpretations of many-worlds using einselection to explain how a small number of classical pointer states can emerge from the enormous Hilbert space of superpositions have been proposed by Wojciech H. Zurek.

space of states of the form

Another version of Hahn – Banach theorem states that if V is a vector space over the scalar field K ( either the real numbers R or the complex numbers C ), if is a seminorm, and is a K-linear functional on a K-linear subspace U of V which is dominated by on U in absolute value,

The holographic principle is a property of quantum gravity and string theories which states that the description of a volume of space can be thought of as encoded on a boundary to the region — preferably a light-like boundary like a gravitational horizon.

The 2009 constitution of Bolivia states that the country has an unrenounceable right over the territory that gives it access to the Pacific Ocean and its maritime space.

In other words, symmetric and antisymmetric states are essentially unchanged under the exchange of particle labels: they are only multiplied by a factor of + 1 or − 1, rather than being " rotated " somewhere else in the Hilbert space.

Louis Feldman, who believes the Josephus passage on John is authentic, states that Christian interpolators would have been very unlikely to have devoted almost twice as much space to John ( 163 words ) as to Jesus ( 89 words ).

An analog joystick is a joystick which has continuous states, i. e. returns an angle measure of the movement in any direction in the plane or the space ( usually using potentiometers ), whereas a digital joystick gives only on / off signals for four different directions and its mechanically possible combinations ( such as up-right or down-left ).

This states that every Hausdorff second-countable regular space is metrizable.

Although the FAI initially refused to recognize Gagarin's pioneering flight, it subsequently amended its position to recognize ( as the FAI website states ) that he was the " first human being to journey into outer space ".

A classical description can be given in a fairly direct way by a phase space model of mechanics: states are points in a symplectic phase space, observables are real-valued functions on it, time evolution is given by a one-parameter group of symplectic transformations of the phase space, and physical symmetries are realized by symplectic transformations.