Cavagnaro, Haight - Dictionary of Classical and Theoretical Mathematics (523108), страница 20
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Also calleda ray. See also closed half line.Hausdorff metricLet (X, d) be a metricspace. If A ⊂ X and > 0, let U (A, ) bethe -neighborhood of A. That is,U (A, ) = ∪a∈A B(a, )where B(a, ) = {x ∈ X : d(x, a) < }.Let H denote the collection of all (non-empty)closed, bounded subsets of X. If A, B ∈ Hthen the Hausdorff Metric on H is defined byD(A, B)= inf{ : A ⊂ U (A, ) and B ⊂ U (B, )} .half plane(1) (Open) One of the two connected sets remaining after deleting a line froma plane.(2) (Closed) An open half plane, togetherwith the deleted line. Example: On the Cartesian plane an open half plane is the set {(x, y) :ax + by + c > 0} where (a, b) = (0, 0), a and care constant, the corresponding closed half planeis the set {(x, y) : ax + by + c ≥ 0}.half spaceOne of the two connected setsremaining after deleting from a sphere its intersection with a plane through the center.halting problem Informally, the halting problem asks if there is an effective procedure which,given an arbitrary effective procedure and inputa natural number n, answers “yes” if that program on input n halts, and outputs “no” otherwise.Formally, the halting set K0 is the set of codesof pairs of natural numbers (e, x) such that thepartial recursive function with Gödel numbere is defined on input x; i.e., K0 = {e, x :ϕe (x) is defined}, where ϕe is the partial recursive function with Gödel number e.
The halting set is not recursive (computable), as a setof natural numbers, although it is recursively(computably) enumerable. Another halting setis K = {e : ϕe (e) is defined}. The set K is alsocomputably (recursively) enumerable but non–computable (non–recursive).Hausdorff Maximal PrincipleEvery chainin a partially ordered set can be extended to amaximal chain. This principle is provable inZFC, and it is provably equivalent to the Axiomof Choice and Zorn’s Lemma in ZF.© 2001 by CRC Press LLCHausdorff topological spaceA topologicalspace X such that, for each pair of distinct pointsx, y ∈ X, there exist open neighborhoods U andV of x and y, respectively, such that U ∩V = ∅.Also called T2 -space.
See separation axioms.height (of a tree)The least ordinal α suchthat the αth level of a given tree T is empty. Thatis, α is the first ordinal for which there is no element in T whose predecessorshave ordertypeα. For example, if T = {a}, {a, b} orderedby inclusion, then Lev0 (T ) contains the set {a},Lev1 (T ) contains {a, b}, and the height of T istwo.Equivalently, the height of T may be definedbyheight(T ) = sup ordertype({s ∈ T : s < t}) + 1 .t∈Theptagon A plane polygon with seven sides.A (convex) heptagon is called regular when itssides have equal length. In that case, its vertices lie on a circle and all of the edges joiningtwo neighboring ones are of equal length; forexample, the vertices cos 2π7 i , sin 2π7 i , i =0, .
. . , 6.hereditary property of a topological spaceA property P of a topological space X such thatevery subspace A ⊆ X has property P. For example, the property of being Hausdorff is hereditary, while the property of being compact is not.hexagonA plane polygon with six vertices.hexahedronA polyhedron with six faces.The most familiar regular hexahedron is the cube,Hilbert cuberegular for a (convex) polyhedron meaning thatall the faces are equal regular polygons and allthe vertices belong to the same number of faces.∞Hilbert cube The Cartesian product n=1 Iof countably manyclosed unit intervals. It is1homeomorphic to ∞n=1 0, n as well as thesubspaceH = {(xn ) ∈ R∞ :∞xn2 < ∞} .n=1holomorphic functionA functionf (z1 , .
. . , zn )that is equal to the sum of an absolutely convergent power series in a suitable polydisc neareach point of its domain (the radius may dependon the point):f (z1 , . . . , zn ) =c(a1 , . . . , an )z1a1 . . . znan ,aj ≥0c(a1 , . . . , an ) ∈ C.
When n = 1, this conditionis equivalent to the Cauchy-Riemann equations:∂u∂v ∂v∂v=,=−∂x∂y ∂y∂xfor z = x + iy, f (z) = u(x, y) + iv(x, y).Examples include polynomials in z and the exponential function ez .holomorphic local coordinate systemFora complex analytic manifold of dimension n, abiholomorphic identification φU (p) of a suitableopen neighborhood U (p) of each point p withthe open ball of radius 1 and center the origin inCn , φU (p) : B0 → U (p).homogeneous topological space A topological space X such that, for each pair of points xand y in X, there is a homeomorphism h : X →X such that h(x) = y.homology class of a mapLet f : X → Ybe continuous and let denote the homotopyequivalence relation.
The homotopy class of fis the equivalence class[f ] = {g : X → Y : g is continuous and g f } .© 2001 by CRC Press LLCIf A ⊆ X is a subspace, then the homotopyclass of f rel A is the equivalence class [f ]A= {g : X → Y : g is continuous and g f rel A} .homology equivalence Let T denote the category of all pairs (X, A) of topological spaces,where X is identified with the pair (X, ∅).
Ahomotopy equivalence between the pairs (X, A)and (Y, B) is a pair of functions f : (X, A) →(Y, B) and g : (Y, B) → (X, A), such that g ◦ fis homotopic to the identity map iX of X andf ◦ g is homotopic to the identity map iY of Y .homology groupLet n be a positive integer, X a topological space, and x a point inX. Then the nth homotopy group πn (X, x) isdefined to be the group of homotopy classesof maps of the standard sphere S n to X taking a fixed base point ∗ of S n to x.
In thecase n = 1, this is the fundamental group ofX (with base point x), with concatenation ofloops inducing the group operation. In dimensions higher than 1, the group operation turnsout to be commutative. Another way of definingπn (X, x) is to take homotopy classes of maps of[0, 1]n to X taking the boundary to x. With thisdefinition, it is easier to define the group operation by concatenation: f + g : [0, 1]n −→X is defined by: f + g(t1 , t2 , .
. . , tn ) equalsf (2t1 , t2 , . . . , tn ) for 0 ≤ t ≤ 21 , and it equalsg(2t1 − 1, t2 , . . . , tn ) for 21 ≤ t1 ≤ 1.homology theoryA homology theory on acategory T of pairs of topological spaces (X, A)consists of(i.) a functor Hp from T to the categoryof Abelian groups A for each integer p ≥ 0,where the image of the pair (X, A) is denotedby Hp (X, A) and(ii.) a natural transformation∂p : Hp (X, A) → Hp−1 (A)for each integer p ≥ 0, where A denotes the pair(A, ∅), which satisfy the Eilenberg-Steenrod Axioms.
See Eilenberg-Steenrod Axioms.homothety A transformation of the Euclideanplane to itself which takes every triangle to asimilar triangle. The homotheties of the planehypotenuseform a group under composition, which contains all isometries and the dilations given byf (x, y) = (ax, ay), a = 0.homotopy type of a spaceLet T denotethe category of all pairs (X, A) of topologicalspaces, where X is identified with the pair (X, ∅).The homotopy type of the pair (X, A) is theequivalence class[(X, A)] = {(Y, B) : (Y, B)is homotopy equivalent to (X,A)} .Hopf bundleThe bundle S 1 −→ S 3 −→S 2 formed as follows. Consider S 3 as the unitsphere in C2 (where C denotes the complexnumbers).
The sphere S 2 is given by CP1 , thespace of complex lines in C2 , by identifyingthe line through the point (z1 , z2 ) with the pointz2 /z1 in C when z1 = 0 and identifying C withthe sphere less one point; the complex line givenby points (0, z2 ) is identified with the remainingpoint on S 2 .
The projection S 3 −→ S 2 is givenby the map which sends (z1 , z2 ) to the complexline through (z1 , z2 ) (thought of as a point ofCP1 identified with a point in S 2 ). Each fiber ishomeomorphic to S 1 .Using quaternions and Cayley numbers instead of complex numbers, one can define analogous bundles S 7 −→ S 4 and S 15 −→ S 8 withfibers S 3 and S 7 , respectively. All three bundlesare called Hopf bundles.hyperbolic paraboloid One of the quadraticsurfaces in R3 . Since a symmetric matrix overthe reals is congruent to one in diagonal form,the (non-degenerate) quadric surfaces are classified, by the sign of their eigenvalues and theconfiguration of their points at infinity into ellipsoids, hyperboloids, elliptic paraboloids, andhyperbolic paraboloids, the latter having canony2x2ical equation 2zc = a 2 − b2 for some non-zeroconstants a, b, c.hyperbolic plane A plane satisfying the axioms of hyperbolic geometry, which compriseHilbert’s axioms of plane geometry and the “characteristic axiom of hyperbolic geometry”: for© 2001 by CRC Press LLCany line l and point p not on l, there are atleast two lines on p not meeting l.
A modelof the hyperbolic plane is given by the unit discD = {(x, y) ∈ R2 : x 2 + y 2 < 1}, with thedx 2 +dy 2Poincaré metric ds 2 = (1−(x2 +y 2 )/4)2 , the linesbeing geodesic.hyperboloidOne of the quadric surfaces inR3 , like the hyperbolic paraboloid. See hyperbolic paraboloid.
The canonical equation for the222hyperboloid of one sheet is xa 2 − yb2 + cz2 = 1and that for the hyperboloid of two sheets is22x2− yb2 − cz2 = 1, where a, b, c are non-zeroa2constants. As the name suggests, one differencebetween the two cases is that one surface consists of one connected component, the other oftwo.hyperelliptic surface A Riemann surface X,namely a compact complex manifold of dimension 1, which generalizes the complex torus, inthe following sense: there exists a holomorphicmap X → P1 of degree 2, with 2g + 2 branchpoints where 2 − 2g is the Euler characteristicof the topological surface X.
See compact complex manifold, complex torus. The number gis called the genus of the surface. Also calledelliptic curve, in the case of genus 1.hyperplane In n-dimensional (affine or projective) space of dimension n, a subspace of dimension n − 1.hyperplane at infinity In n-dimensional projective space Pn , with given coordinate system(x0 : x1 : . . . : xn ), a hyperplane H , typicallygiven by the equation x0 = 0. The complementPn \H can thus be identified with affine n-spaceAn with coordinates ( xx01 , . .
. xxn0 ).hypersurface In affine or projective space, asubset defined by one (nonzero) algebraic equation in the coordinates. A hyperplane is an example of hypersurface, which is defined by onelinear equation. See also hyperplane.hypotenuseThe side of a right triangle opposite the right angle. It is the longest side ofthe triangle.inaccessible cardinal (weakly)f isf (C)IicosahedronA polyhedron with 20 faces.The icosahedron is one of the five (convex) polyhedra that can be regular.idealLet S be a nonempty set and let P(S)be the power set of S. A set I ⊆ P(S) is anideal on S if(i.) ∅ ∈ I(ii.) for all X, Y ∈ I , X ∪ Y ∈ I ,(iii.) for all X, Y , if X ∈ I and Y ⊆ X, thenY ∈ I.As an example, let S be the set N of naturalnumbers and let I be the set of all finite subsetsof N.
Then I is an ideal on S.identification map A continuous onto mapping f : X → Y such that the topology on Y isthe identification topology; that is, U is open inY if and only if f −1 (U ) is open in X. See alsoquotient map.identification space An identification spaceof a topological space X is a set Y endowedwith the topology induced by an onto mappingf : X → Y .
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