mathematics, a Borel isomorphism is a measurable bijective function between two standard Borel spaces. By Souslin's theorem in standard Borel spaces (which...

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...

unique, up to isomorphism of measurable spaces. A measurable space ( X , Σ ) {\displaystyle (X,\Sigma )} is said to be "standard Borel" if there exists...

two uncountable Polish spaces, there is a Borel isomorphism; that is, a bijection that preserves the Borel structure. In particular, every uncountable...

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 ∖...

topology, Borel−Moore homology or homology with closed support is a homology theory for locally compact spaces, introduced by Armand Borel and John Moore...

_{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...

{\mathcal {R}}} is not a Borel set. There are a variety of other anti-classification results. For example, replacing isomorphism with Kakutani equivalence...

Frédéric; Fasel, Jean; Jin, Fangzhou; Khan, Adeel (2019-02-06). "Borel isomorphism and absolute purity". arXiv:1902.02055 [math.AG]. Fujiwara, K.: A...

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)...

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...

In mathematics, equivariant cohomology (or Borel cohomology) is a cohomology theory from algebraic topology which applies to topological spaces with a...

space. More formally, the third level classifies spaces up to isomorphism. An isomorphism between two spaces is defined as a one-to-one correspondence...

geometry, and dynamics to questions such as the Borel conjecture. The Farrell-Jones conjecture implies the Borel Conjecture for manifolds of dimension greater...

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...

isomorphism, introduced by Harish-Chandra (1951), is an isomorphism of commutative rings constructed in the theory of Lie algebras. The isomorphism maps...

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...

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...

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)...

the fibered isomorphism conjecture with respect to the family F {\displaystyle F} if and only if it satisfies the (fibered) isomorphism conjecture with...

Wallach 1998, Olver 1999, Sharpe 1997. Borel & Casselman 1979, Gelbart 1984. See the previous footnotes and also Borel (2001). Simson, Skowronski & Assem...

a field, Poincaré duality is naturally formulated as an isomorphism from cohomology to Borel–Moore homology. Verdier duality is a vast generalization...

countable Borel equivalence relation. The isomorphism equivalence relation between various classes of models, while not being countable Borel equivalence...

\mathbb {C} } -linear isomorphism. Conversely,[clarification needed] suppose there is a C {\displaystyle \mathbb {C} } -linear isomorphism τ : g → ∼ g ¯ {\displaystyle...

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...

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)...

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...

Borel 1991, 1.6(2), p. 49. Borel 1991, Corollary 1.4, p. 47. Borel 1991, Theorem 6.8, p. 98. Borel 1991, 3.5, p. 65. Borel 1991, pp. 55-56. Borel 1991...

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...

{\displaystyle \mathbb {R} ^{2},} ⁠ also forms a quadratic space. The ring isomorphism D → R 2 x + y j ↦ ( x − y , x + y ) {\displaystyle {\begin{aligned}D&\to...