Definition & Betydelse | Engelska ordet HOMEOMORPHISMS

Definition av HOMEOMORPHISMS

1. böjningsform av homeomorphism

Exempel på hur man kan använda HOMEOMORPHISMS i en mening

• One consequence of the Banach fixed-point theorem is that small Lipschitz perturbations of the identity are bi-lipschitz homeomorphisms.
• The category LH whose objects are topological spaces and whose morphisms are local homeomorphisms is locally Cartesian closed, since LH/X is equivalent to the category of sheaves.
• The group F also has realizations in terms of operations on ordered rooted binary trees, and as a subgroup of the piecewise linear homeomorphisms of the unit interval that preserve orientation and whose non-differentiable points are dyadic rationals and whose slopes are all powers of 2.
• We can consider the set of homeomorphisms from the space into itself, that is, continuous maps with continuous inverses: functions which stretch and deform the space continuously without breaking or gluing the space.
• However, unlike those for a group, the axioms defining a pseudogroup are not purely algebraic; the further requirements are related to the possibility of restricting and of patching homeomorphisms (similar to the gluing axiom for sections of a sheaf).
• In some applications, particularly surfaces, the homeomorphism group is studied via this short exact sequence, and by first studying the mapping class group and group of isotopically trivial homeomorphisms, and then (at times) the extension.
• Thurston's classification applies to homeomorphisms of orientable surfaces of genus ≥ 2, but the type of a homeomorphism only depends on its associated element of the mapping class group Mod(S).
• This is the fibered part of William Thurston's geometrization theorem for Haken manifolds, whose proof requires the Nielsen–Thurston classification for surface homeomorphisms as well as deep results in the theory of Kleinian groups.
• The forgetful functor from topological spaces to sets is not monadic as it does not reflect isomorphisms: continuous bijections between (non-compact or non-Hausdorff) topological spaces need not be homeomorphisms.
• In the case of the unit disk, Teichmüller theory implies that the homomorphism carries quasiconformal homeomorphisms of the disk onto the group of quasi-Möbius homeomorphisms of the circle (using for example the Ahlfors–Beurling or Douady–Earle extension): it follows that the homomorphism from the quasi-isometry group into the quasi-Möbius group is surjective.
• The existence of such counter-examples that are homeomorphisms was shown in 1943 by Shizuo Kakutani, who may have first written down a proof of the contractibility of the unit sphere.
• Alternatively, many advanced methods for spatial normalization are building on structure preserving transformations homeomorphisms and diffeomorphisms since they carry smooth submanifolds smoothly during transformation.
• This conjecture is false if topological manifolds and homeomorphisms are replaced by smooth manifolds and diffeomorphisms; counterexamples can be constructed by taking a connected sum with an exotic sphere.
• In a subsequent paper Bestvina and Handel applied the train track techniques to obtain an effective proof of Thurston's classification of homeomorphisms of compact surfaces (with or without boundary) which says that every such homeomorphism is, up to isotopy, is either reducible, of finite order or pseudo-anosov.
• An extension for quasisymmetric homeomorphisms had previously been given by Lars Ahlfors and Arne Beurling; a different equivariant construction had been given in 1985 by Pekka Tukia.
• Kruskal's tree theorem states that, in every infinite set of finite trees, there exists a pair of trees one of which is homeomorphically embedded into the other; another way of stating the same fact is that the homeomorphisms of trees form a well-quasi-ordering.

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