Definition & Betydelse | Engelska ordet HOMEOMORPHISM

Definition av HOMEOMORPHISM

1. (matematik) homeomorfi

Exempel på hur man kan använda HOMEOMORPHISM i en mening

• In mathematics and more specifically in topology, a homeomorphism (from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function between topological spaces that has a continuous inverse function.
• The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.
• If X is a Tychonoff space then the map from X to its image in βX is a homeomorphism, so X can be thought of as a (dense) subspace of βX; every other compact Hausdorff space that densely contains X is a quotient of βX.
• If a continuous bijection has as its domain a compact space and its codomain is Hausdorff, then it is a homeomorphism.
• So importantly, for a Banach space to be reflexive, it is not enough for it to be isometrically isomorphic to its bidual; it is the canonical evaluation map in particular that has to be a homeomorphism.
• The case of an algebraically closed ground field is especially important in algebraic geometry, because, in this case, the homeomorphism above is a map between the affine space and the set of all maximal ideals of the ring of functions (this is Hilbert's Nullstellensatz).
• an acyclic, 3-dimensional continuum which admits a fixed point free homeomorphism onto itself; also 2-dimensional, contractible polyhedra which have no free edge.
• A special case of this is when the group G in question is the automorphism group of the space X – here "automorphism group" can mean isometry group, diffeomorphism group, or homeomorphism group.
• Cobordism is a much coarser equivalence relation than diffeomorphism or homeomorphism of manifolds, and is significantly easier to study and compute.
• canonically identifies these two Cartesian products; moreover, this map is a homeomorphism when these products are endowed with their product topologies.
• Sobriety of X is precisely a condition that forces the lattice of open subsets of X to determine X up to homeomorphism, which is relevant to pointless topology.
• Ambient isotopy (or h-isotopy), two subsets of a fixed topological space are ambient isotopic if there is a homeomorphism, isotopic to the identity map of the ambient space, which carries one subset to the other.
• In particular, if the map is a continuous bijection (a homeomorphism), so that the two spaces have the same topology, then their -th homotopy groups are isomorphic for all.
• 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.
• In this view, embeddings of graphs into a surface or as subdivisions of other graphs are both instances of topological embedding, homeomorphism of graphs is just the specialization of topological homeomorphism, the notion of a connected graph coincides with topological connectedness, and a connected graph is a tree if and only if its fundamental group is trivial.
• The three types in this classification are not mutually exclusive, though a pseudo-Anosov homeomorphism is never periodic or reducible.
• A quasiconformal mapping between two Riemann surfaces is a homeomorphism which deforms the conformal structure in a bounded manner over the surface.
• A homeomorphism between Baire space and the irrationals can be constructed using continued fractions.
• Since the previously described construction results in a class of 4-manifolds that are homeomorphic if and only if their groups are isomorphic, the homeomorphism problem for 4-manifolds is undecidable.
• This homeomorphism is essentially that of currying, modulo the quotients needed to convert the products to reduced products.

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