mathematics, a Borel isomorphism is a measurable bijective function between two standard Borel spaces. By Souslin's theorem in standard Borel spaces (which...
2 KB (200 words) - 00:25, 9 January 2023
isomorphic. A standard Borel space is the Borel space associated to a Polish space. A standard Borel space is characterized up to isomorphism by its cardinality...
13 KB (1,793 words) - 02:48, 4 June 2024
unique, up to isomorphism of measurable spaces. A measurable space ( X , Σ ) {\displaystyle (X,\Sigma )} is said to be "standard Borel" if there exists...
3 KB (423 words) - 07:18, 27 May 2024
measures on standard Borel spaces X and Y respectively. Then there is a μ null subset N of X, a ν null subset M of Y and a Borel isomorphism ϕ : X ∖ N → Y ∖...
10 KB (1,547 words) - 15:36, 16 February 2024
two uncountable Polish spaces, there is a Borel isomorphism; that is, a bijection that preserves the Borel structure. In particular, every uncountable...
12 KB (1,494 words) - 20:32, 8 September 2024
topology, Borel−Moore homology or homology with closed support is a homology theory for locally compact spaces, introduced by Armand Borel and John Moore...
14 KB (2,666 words) - 13:39, 22 July 2024
_{Y}^{\oplus }K_{y}d\nu (y)} and μ, ν are standard measures, then there is a Borel isomorphism φ : X − E → Y − F {\displaystyle \varphi :X-E\rightarrow Y-F} where...
17 KB (2,820 words) - 03:52, 5 March 2024
Measure-preserving dynamical system (redirect from Isomorphism of dynamical systems)
{\mathcal {R}}} is not a Borel set. There are a variety of other anti-classification results. For example, replacing isomorphism with Kakutani equivalence...
23 KB (3,592 words) - 13:02, 9 August 2024
Frédéric; Fasel, Jean; Jin, Fangzhou; Khan, Adeel (2019-02-06). "Borel isomorphism and absolute purity". arXiv:1902.02055 [math.AG]. Fujiwara, K.: A...
2 KB (256 words) - 13:25, 20 December 2023
the inverse of that restriction is a Borel section of f—it is a Borel isomorphism. Uniformization Hahn–Banach theorem Section 4 of Parthasarathy (1967)...
2 KB (201 words) - 06:38, 13 December 2021
Equivariant cohomology (redirect from Borel construction)
In mathematics, equivariant cohomology (or Borel cohomology) is a cohomology theory from algebraic topology which applies to topological spaces with a...
12 KB (1,813 words) - 12:29, 30 April 2024
isomorphism, introduced by Harish-Chandra (1951), is an isomorphism of commutative rings constructed in the theory of Lie algebras. The isomorphism maps...
16 KB (2,545 words) - 18:47, 26 January 2024
Poincaré duality (section Thom isomorphism formulation)
the isomorphism remains valid over any coefficient ring. In the case where an oriented manifold is not compact, one has to replace homology by Borel–Moore...
17 KB (2,694 words) - 02:09, 28 September 2024
outlined here (in fact, the result is a coalgebra isomorphism, and not merely a K-module isomorphism, equipping both S(L) and U(L) with their natural coalgebra...
14 KB (1,922 words) - 07:37, 10 June 2024
space. More formally, the third level classifies spaces up to isomorphism. An isomorphism between two spaces is defined as a one-to-one correspondence...
69 KB (9,326 words) - 21:09, 27 August 2024
atomless. The complete Cantor algebra is the complete Boolean algebra of Borel subsets of the reals modulo meager sets (Balcar & Jech 2006). It is isomorphic...
2 KB (204 words) - 18:02, 23 June 2020
isomorphic (in the category of Borel spaces) to the underlying Borel space of a complete separable metric space. Mackey called Borel spaces with this property...
12 KB (1,753 words) - 20:34, 24 January 2024
a field, Poincaré duality is naturally formulated as an isomorphism from cohomology to Borel–Moore homology. Verdier duality is a vast generalization...
36 KB (5,832 words) - 15:01, 25 July 2024
Wallach 1998, Olver 1999, Sharpe 1997. Borel & Casselman 1979, Gelbart 1984. See the previous footnotes and also Borel (2001). Simson, Skowronski & Assem...
55 KB (7,184 words) - 17:41, 8 July 2024
a Hilbert space, but the same notion serves to define the concept of isomorphism between Hilbert spaces. Definition 1. A unitary operator is a bounded...
9 KB (1,264 words) - 15:09, 5 September 2024
1.8. Borel (1991), section 23.4. Borel (1991), section 23.2. Borel & Tits (1971), Corollaire 3.8. Platonov & Rapinchuk (1994), Theorem 3.1. Borel (1991)...
55 KB (7,845 words) - 18:28, 24 April 2024
Linear algebraic group (section Borel subgroups)
f: Gm → Gm defined by x ↦ xp induces an isomorphism of abstract groups k* → k*, but f is not an isomorphism of algebraic groups (because x1/p is not...
41 KB (6,000 words) - 20:12, 10 January 2024
the fibered isomorphism conjecture with respect to the family F {\displaystyle F} if and only if it satisfies the (fibered) isomorphism conjecture with...
12 KB (2,216 words) - 14:16, 25 March 2024
countable Borel equivalence relation. The isomorphism equivalence relation between various classes of models, while not being countable Borel equivalence...
11 KB (1,818 words) - 12:42, 11 August 2024
{\text{End}}({\widehat {G}})^{\text{op}}} . More categorically, this is not just an isomorphism of endomorphism algebras, but a contravariant equivalence of categories...
39 KB (5,806 words) - 03:42, 8 May 2024
\mathbb {C} } -linear isomorphism. Conversely,[clarification needed] suppose there is a C {\displaystyle \mathbb {C} } -linear isomorphism τ : g → ∼ g ¯ {\displaystyle...
5 KB (830 words) - 21:27, 9 July 2024
map in L-theory. The Borel conjecture on the rigidity of aspherical manifolds is equivalent to the assembly map being an isomorphism. Davis, James F. (2000)...
5 KB (571 words) - 21:26, 7 June 2024
Lie group (section Homomorphisms and isomorphisms)
Lie algebras; it is then reasonable to ask how isomorphism classes of Lie groups relate to isomorphism classes of Lie algebras. The first result in this...
64 KB (9,479 words) - 09:21, 13 September 2024
Solvable group (section Borel subgroups)
are two of the Borel subgroups. The example given above, the subgroup B {\displaystyle B} in G L 2 {\displaystyle GL_{2}} , is a Borel subgroup. In G...
18 KB (3,073 words) - 09:51, 27 August 2024
y)=(s-h(y),T(y)).} These transformations are related by an invertible Borel isomorphism Φ from R × Ω onto Z × X defined by Φ ( t , y ) = ( N t ( y ) , S t...
36 KB (5,093 words) - 23:31, 26 August 2024