In set theory, a subset of a Polish space X {\displaystyle X} is ∞-Borel if it can be obtained by starting with the open subsets of X {\displaystyle X}...

Descriptive set theory Analytic set Analytical hierarchy Borel equivalence relation Infinity-Borel set Lightface analytic game Perfect set property Polish...

In real analysis the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states: For a subset S of Euclidean space Rn, the following two statements...

Descriptive set theory Analytic set Analytical hierarchy Borel equivalence relation Infinity-Borel set Lightface analytic game Perfect set property Polish...

general topology. Borel set Analytic set C-measurable set Projective set Inductive set Infinity-Borel set Suslin set Homogeneously Suslin set Weakly homogeneously...

hierarchy. Many properties of Borel sets can be established in ZFC, but proving these properties hold for more complicated sets requires additional axioms...

assigns measure 1 to Borel sets containing an unbounded closed subset of the countable ordinals and assigns 0 to other Borel sets is a Borel probability measure...

probabilities of the En diverges to infinity, then the probability that infinitely many of them occur is 1. That is: Second Borel–Cantelli Lemma — If ∑ n = 1...

functions may be vanishing at infinity or have compact support, and the measures can be Baire measures or regular Borel measures or Radon measures or...

today because they are the primary setting for descriptive set theory, including the study of Borel equivalence relations. Polish spaces are also a convenient...

the Russian school of point-set topology, under the direction of Pavel Alexandrov and Pavel Urysohn, formulated Heine–Borel compactness in a way that could...

(often called a point at infinity) and defining the open sets of the new space to be the open sets of X together with the sets of the form G ∪ {∞}, where...

of Borel sets of a Hausdorff topological space X that is finite on all compact sets, outer regular on all Borel sets, and inner regular on open sets. These...

than infinity. In the special case when J(x) = ex this gives one (weak) form of Borel summation. Valiron's method is a generalization of Borel summation...

subsets of G {\displaystyle G} is called the Borel algebra. An element of the Borel algebra is called a Borel set. If g {\displaystyle g} is an element of...

Cantor set is defined this way. If the limit of 1 A n ( x ) , {\displaystyle \mathbb {1} _{A_{n}}(x),} as n {\displaystyle n} goes to infinity, exists...

cardinality and realized that infinite sets can have different cardinalities. The aleph numbers differ from the infinity (∞) commonly found in algebra and...

the use of the "monkey metaphor" is that of French mathematician Émile Borel in 1913, but the first instance may have been even earlier. Jorge Luis Borges...

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

set of all open sets in R n {\displaystyle \mathbb {R} ^{n}} ) the Borel σ-algebra on R {\displaystyle \mathbb {R} } (i.e. the set of all Borel sets in...

also contributed to refinement of the law, including Chebyshev, Markov, Borel, Cantelli, Kolmogorov and Khinchin. Markov showed that the law can apply...

ground at infinity for the harmonic or Newtonian capacity, and with respect to a surface for the condenser capacity. The notion of capacity of a set and of...

particularly complex dynamics, the escaping set of an entire function ƒ consists of all points that tend to infinity under the repeated application of ƒ. That...

Extended real number line Dedekind cut 0 1 0.999... Infinity Open set Neighbourhood Cantor set Derived set (mathematics) Completeness Limit superior and limit...

Cantor set; the Hilbert cube. The Euclidean spaces Rn (and in particular the real line R) are locally compact as a consequence of the Heine–Borel theorem...

Émile Borel, Henri Lebesgue, Nikolai Luzin, Johann Radon, Constantin Carathéodory, and Maurice Fréchet, among others. Let X {\displaystyle X} be a set and...

the maximal product measure a set has measure infinity unless it is contained in the union of a countable number of sets of the form A×B, where either...

set. Borel determinacy is implied by the existence of all beths of countable index. Transfinite number Uncountable set Jech, Thomas (2002). Set Theory...

equals ∅ {\displaystyle \emptyset } , is a set. NBG's axiom of infinity is implied by ZFC's axiom of infinity: ∃ a [ ∅ ∈ a ∧ ∀ x ( x ∈ a ⟹ x ∪ { x } ∈ a...

that the Lebesgue measure of the unit cube [0,1]d is 1. In fact, for any Borel set E, λ d ( E ) = 2 − d α d H d ( E ) , {\displaystyle \lambda _{d}(E)=2^{-d}\alpha...