Cavagnaro, Haight - Dictionary of Classical and Theoretical Mathematics (523108), страница 39
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As a filter, U must benonempty, closed under ∧, not contain 0, andbe closed upwards: for all u ∈ U and b ∈ B, ifu ≤ b then b ∈ U. The maximality condition isequivalent to requiring that for all b ∈ B, eitherb ∈ U or ¬b ∈ U.Any filter can be extended to an ultrafilter,and, using a weak form of the Axiom of Choice,any subset of a Boolean algebra with the finiteintersection property can be extended to an ultrafilter.ultrapowerAn ultrapower of an L-structure A is a reduced product U A, where U isan ultrafilter over the index set I .
The reducedproduct is formed by declaring, for x and y inthe Cartesian product I A, that x ≡U y if andonly if the set of coordinates where x and y agreeis in the ultrafilter U:{i ∈ I : x(i) = y(i)} ∈ U .The reduced product U A is then the set of allequivalence classes under ≡U .The fundamental propertyof ultrapowers isthat, for any L-sentence φ, U A |= φ if andonly if {i ∈ I : A |= φ} ∈ U. But because Uis an ultrafilter, ∅ ∈/ U and I ∈ U, and so, theultrapower models φ if and only if the originalstructure A models φ.
Thus, U A ≡ A.See also ultraproduct.ultraproduct An ultraproduct of a set of Lstructures {Ai : i ∈ I } is a reduced productU Ai , where U is an ultrafilter over the indexset I . See ultrapower.The fundamental propertyof ultraproducts isthat for any L-sentence φ, U Ai |= φ if andonly if {i ∈ I : Ai |= φ} ∈ U.umbilical point Let M be a surface in R3 , andlet k1 ≥ k2 be the principal curvature functions.See principal curvature. An umbilical point is© 2001 by CRC Press LLCa point where k1 = k2 .
On the complement ofthe set of umbilical points, the principal curvesform a pair of orthogonal fields of curves onthe surface; the umbilical points are the placeswhere these fields become singular.unbounded set A set of ordinals C ⊆ κ suchthat, for any α < κ, there is a β with α ≤ β < κand β ∈ C. See also closed set, stationary set.uncountable A set that is infinite but not denumerable. For example, R and C are uncountable sets.undecidableA set of objects of some sort,which it is not decidable. See decidable.uniformly continuous functionA functionf : R → R such that, for any > 0, there isa δ > 0 such that for x and x in R, |f (x) −f (x )| < whenever |x − x | < δ. Any continuous f : [a, b] → R is uniformly continuous.More generally, a function f from one metricspace (X, dX ) to another (Y, dY ) is uniformlycontinuous if for any > 0, there is a δ > 0 suchthat, for all x and x in X, dY (f (x), f (x )) < whenever dX (x, x ) < δ.
If X is compact,then any continuous f : X → Y is uniformlycontinuous.Further generalization of the notion is possible in a uniform space. See uniform space.uniform space A set X with the topology induced by a uniformity U. Informally, a uniformity is a way of capturing closeness in a topological space without a metric; that is, it provides a generalization of a metric. Formally, anonempty collection U of subsets of X × X isa uniformity if it satisfies the following conditions:(i.) for all U ∈ U, ⊆ U , where ={(x, x) : x ∈ X} is the diagonal of X;(ii.) for all U ∈ U, U −1 ∈ U, where U −1 ={(y, x) : (x, y) ∈ U };(iii.) for all U and V in U, U ∩ V ∈ U;(iv.) for each U ∈ U there is a V ∈ U withV ◦ V ⊆ U , whereV ◦V =(x, z) : ∃y ∈ X (x, y) ∈ V and (y, z) ∈ V ;uniform topologyand(v.) for all U ∈ U, if U ⊆ V , then V ∈ U.The idea is that x and y will be considered U close to each other if (x, y) ∈ U . Then, forexample, condition (i.) states that x is alwaysU -close to itself.A uniformity U generates a topology on X(the uniform topology) by considering the setsU [x] = {y : (x, y) ∈ U } as basic open sets foreach U ∈ U and x ∈ X.uniform topology(1) See uniform space.(2) The uniform topology on Rα is the topology induced by the bounded sup metricδ(x, y) = sup{min{|xβ − yβ |, 1} : β < α} .This topology is the same as the product topology if α is finite; if α is infinite, the uniformtopology refines the product topology.union (1) The union of any set X, denoted by∪X, is the set whose elements are the membersof the members of X.
That is, a ∈ ∪X if andonly if there exists S ∈ X such that a ∈ S. Forexample, ∪{(0, k) : k ∈ Z} = R+ . If X is anindexed family of sets {Sα : α ∈ I }, where I issome index set, the union of X is often denotedby α∈I Sα .(2) The union of sets A and B, denoted byA ∪ B, is the set of all elements that belong toat least one of A and B. This is a special caseof the previous definition, as A ∪ B = ∪{A, B}.For example, {3, 10} ∪ {3, 5} = {3, 10, 5} andN ∪ R = R. See also Axiom of Union.unit functionThe arithmetic function, denoted u, which returns the value 1 for all positiveintegers, i.e., u(n) = 1 for all integers n ≥ 1.(See arithmetic function.) It is completely (andstrongly) multiplicative.universal bundleA bundle EG −→ BGwith fiber G is a universal bundle with structure group G if EG is contractible and every Gbundle over X is the equivalent to the bundleformed by the pullback of EG −→ BG alongsome map X −→ BG.Example: The universal real line bundle isEO(1) −→ BO(1) equivalent to the covering ofBO(1) = RP∞ (infinite dimensional real pro-© 2001 by CRC Press LLCjective space) by S ∞ , the union over all n ofspheres S n , under the action of Z/2 = O(1).universal elementIf C is any category, S isthe category of sets and functions, and F : C →S is a functor, a universal element of F is apair (A, B), where A is an object of C and B ∈F (A), such that for every pair (A , B ), whereB ∈ F (A ), there exists a unique morphismf : A → A of C with (F (f ))(B) = B .universal mapping property The notion ofa universal mapping property is not a rigorouslydefined one, as many variations exist.
A common pattern that appears in many instances canbe described as follows. A triple (p, A, A ),where A and A are objects of a category C andp : A → A is a morphism of C, has a universal mapping property if, for every morphismf : X → A of C, there exists a unique morphism f : X → A of C such that f = p ◦ f .In most cases, a universal mapping property isused to define a new object.
A standard exampleof defining a tuple having a universal mappingproperty is the product of objects in a category.See product of objects.universal quantifierSee quantifier.universal sentence A sentence σ of a first order language L which has the form ∀v1 . . . ∀vn α,where α is quantifier-free, for some n ≥ 0.universe of setsThe collection of all sets.In Zermelo-Fraenkel set theory (ZFC), the universe of sets, usually denoted by V , can be expressed by the abbreviation V = α Vα , whereeach Vα is a set from the cumulative hierarchy.It is important to note that this union does notdefine a set in ZFC, rather, the above equation issimply an abbreviation for the following statement which is provable in ZFC: (∀x)(∃α) x ∈Vα .
See also cumulative hierarchy.unordered pairA set with exactly two elements. For example, {3, −5} is an unorderedpair. Compare with ordered pair.upper limit topologySee Sorgenfrey line.Urysohn’s Metrization TheoremUrysohn’s LemmaFor any two disjointclosed subsets A and B of a normal topologicalspace X, there is a continuous f : X → [0, 1]such that f (a) = 0 for every a ∈ A andf (b) = 1 for every b ∈ B. That is, normality implies disjoint closed sets may be separatedby continuous functions. The converse is easier:if f is continuous and separates A and B, thenf −1 ([0, 21 )) and f −1 (( 21 , 1]) are disjoint opensets containing A and B, respectively. Thus,normality is equivalent to separation by continuous functions for Hausdorff spaces.© 2001 by CRC Press LLCUrysohn’s Lemma is a vital part of the proofsof Tietze’s Extension Theorem and Urysohn’sMetrization Theorem.Urysohn’s Metrization Theorem Any regular, second countable topological space is metrizable.
In other words, if X is regular and has acountable basis, then there is a metric that induces the topology on X. The proof relies onUrysohn’s Lemma and imbeds X in the cube[0, 1]ω , which is also separable. See alsoUrysohn’s Lemma.von Mangoldt functionVvalidLet L be a first order language and letα be a well-formed formula of L.
If, for everystructure A for L and for every s : V → A, Asatisfies α with s, then α is valid or is a validity.(Here, V is the set of variables of L and A is theuniverse of A.)As an example, let L be the language of equality, =. The formula(v1 = v2 ∧ v2 = v3 ) → v1 = v3is valid.validitySee valid.Venn diagramA schematic device used toverify relations among sets contained within auniversal set U .The universal set U may be represented by aclosed figure such as a rectangle. A set A ⊂ U isthen represented by the interior of some closedregion within U , while the statement x ∈ Ais indicated as a point within the region A. Therelation A ⊂ B is depicted by placing the regionrepresenting A within that of B.The union A ∪ B of two sets may be represented by shading the combined regions including both A and B.
The intersection A ∩ B isindicated by shading the overlapping portionsof the regions A and B and the complement ofA or A is indicated by shading the region withinU which is outside A.The relation (A ∪ B) = A ∩ B is shown inthe figure. The top diagram indicates by shadingthe set (A∪B) and the bottom diagram indicatesthe common elements of A and B .von Mangoldt functiontion.© 2001 by CRC Press LLCSee Mangoldt func-Top:(A ∪ B) . Bottom: A ∩ B well-orderingWWang exact sequenceLet F −→ E −→S n be a fiber bundle with n ≥ 2 and F pathconnected. Then there is a long exact sequence· · · −→ H k (E) −→ H k (F ) −→ H k−n+1 (F )−→ H k+1 (E) −→ · · ·called the Wang exact sequence.This sequence is derived from the spectralsequence for the fiber bundle, which in this casehas only one non-trivial differential.
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