Cavagnaro, Haight - Dictionary of Classical and Theoretical Mathematics (523108), страница 21
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This topology (the identificationtopology) is given by: U ⊆ Y is open if andonly if f −1 (U ) is open in X. See also quotientspace.identification topologyspace.See identificationidentity functionThe arithmetic function,denoted I , which has the value 1 when n = 1 andhas the value 0 when n > 1, i.e., I (n) = n1 ,the floor function applied to n1 . It is completely(and strongly) multiplicative. This function isthe identity under Dirichlet multiplication. Seearithmetic function, Dirichlet multiplication.imageLet f : A → B be a function, andlet x ∈ A. The image of x under f is f (x), theunique element of B to which x is mapped byf . Given a subset C ⊆ A, the image of C under© 2001 by CRC Press LLC=={f (x) : x ∈ C}{y ∈ B : (∃x ∈ C)[f (x) = y]}.imaginary axisThe y-axis, which corresponds to the purely imaginary numbers in theArgand diagram for the complex numbers,namely the identification of x + iy ∈ C withthe point (x, y) ∈ R2 .imbedding A one-to-one continuous map f :X → Y , between topological spaces, for whichthe restriction f ∗ : X → f (X) to its range is ahomeomorphism.
Heref (X) = y ∈ Y : ∃x ∈ X (f (x) = y)and f ∗ is required not only to be one-to-one,onto, and continuous, but to have a continuousinverse.immersed submanifoldThe image f (M)of an immersion f : M → N between twomanifolds. Each point in M has a neighborhoodon which f is an embedding. However, the mapf need not be an embedding, so f (M) need notbe a manifold with the induced topology as asubset of N , even if f is globally 1-1. A simpleexample is given by the immersion of the openinterval into the plane whose image is the figure6.
A more complicated example is given byviewing the torus as the quotient of the planeby the integer lattice. Then a line of irrationalslope is mapped onto a dense subset of the torus,which is an immersed submanifold.inaccessible cardinal (strongly)A cardinalκ which is uncountable, regular, and satisfiesthe condition 2α < κ for all α < κ, i.e., κis a strong limit cardinal. (Any strongly inaccessible cardinal is weakly inaccessible since astrong limit cardinal is a limit cardinal. The existence of strongly inaccessible cardinals cannotbe proved in Zermelo-Fraenkel set theory withthe Axiom of Choice.) See regular cardinal.inaccessible cardinal (weakly)A cardinalwhich is uncountable, a regular cardinal, and alimit cardinal.
(The existence of weakly inaccessible cardinals cannot be proved in ZermeloFraenkel set theory with the Axiom of Choice.incenter of triangleAs seen in the definition of inaccessible cardinal (strongly), every strongly inaccessible cardinal is weakly inaccessible. [See inaccessiblecardinal (strongly).] If one assumes the generalized continuum hypothesis, then the converseis true.) See regular cardinal, limit cardinal.incenter of triangle The center of the uniquecircle which can be inscribed in a given triangle.It is located at the intersection of the internalbisectors of the three vertices of the triangle.incommensurableTwo line segments XYand X Y such that there is no line segment ABwith the property that each of XY and X Y haslength that is an exact (integer) multiple of thelength of AB.
That is, there is no unit of measure with respect to which both segments haveinteger length.For example, the hypotenuse of an isoscelesright triangle and a leg√of the triangle are incommensurable because 2 is an irrational number.incomparable (elements of a partial ordering) If (P , ≤) is a partially ordered set, x, y ∈P are incomparable if neither x ≤ y, nor y ≤ x.incompatible (elements of a partial ordering)See compatible (elements of a partial ordering).inconsistentLet L be a first order languageand let be a set of well-formed formulas of L.The set is inconsistent if there exists a wellformed formula α such that both α and (¬α)are provable from (i.e., both α and (¬α) aretheorems of ).
If is inconsistent, then in factevery formula is a theorem of .inconsistent axiomsA set of axioms suchthat there is a statement A such that both A andits negation are provable from the axioms. Seealso inconsistent. For example, in set theory, theaxioms AC (the Axiom of Choice) and AD (theAxiom of Determinacy) are inconsistent, as theAxiom of Determinacy contradicts the Axiomof Choice.indiscernibleA subset I of a model A is aset of indiscernibles if no first-order formula candistinguish between increasing sequences fromI . More precisely, if < is any linear order on I© 2001 by CRC Press LLCand n ∈ N, then for all a1 < a2 < · · · < an andb1 < b2 < · · · < bn in I , A |= φ(ā) if and onlyif A |= φ(b̄) for all L-formulas φ.inductionOne of two techniques used toprove that a given proposition P is true for allnatural numbers. Let P (n) denote the statement“P is true for the natural number n”.
The principle of weak induction states that if(i.) P (0)(ii.) P (m) implies P (m + 1) for any naturalnumber M,then P (n) for all natural numbers n. The principle of strong induction states that if (i.) and(ii.) P (0), P (1), . . . , P (m) implies P (m + 1)for any natural number m, then P (n) is true forany natural number n. The proof technique inthe principle of strong induction may be generalized to any well-ordered set W , giving theprinciple of transfinite induction.inductive setA set A such that ∅ ∈ A, andfor all sets x, if x ∈ A then x + ∈ A, wherex + = x ∪ {x} is the successor of the set x.infimumLet (X, ≤) be a partially orderedset and suppose that Y ⊆ X.
An element z ∈ Xis an infimum, or greatest lower bound, of Y(denoted inf(Y ) or glb(Y )) if z is a lower boundfor Y and r ≤ z for any other element r whichis a lower bound for Y . See lower bound.infinite continued fraction A continued fraction that is not finite. See continued fraction,finite continued fraction.infinite dimensional projective spaceAspace in projective geometry, which generalizesthe n-dimensional projective space Pn . Over afield k, the points of Pn can be coordinatized by(n + 1)-tuples (x0 : x1 : .
. . : xn ) where xi ∈ kand at least one xi = 0, up to the equivalencerelation (x0 : . . . : xn ) ∼ (λx0 : . . . : λxn ) for0 = λ ∈ k. The classical examples are the realprojective space (k = R) and complex projective space (k = C). The infinite dimensionalprojective space can be constructed as a directlimitlim→ Pn = P∞ ,namely a collection of injections πi : Pi → P∞with the property that πj = πi ◦ ρij , whereinterior angleρij : Pj → Pi is the natural inclusion (x0 : .
. . :xj ) → (x0 : . . . : xj : 0 : . . . : 0) for j < i.tinguishing between countably and uncountablyinfinite sets, for example.infinite Grassmann manifold A Grassmannmanifold is the set of all subspaces of a vectorspace V that have a given dimension k. When Vis a real (or complex) vector space, this set is indeed a real (or complex) manifold. If the dimension of V is n, the dimension of the Grassmannmanifold is k(n − k).
An infinite Grassmannmanifold is a generalization of this object for Vof infinite dimension, but for it to be a manifold,care must be taken that there exist local coordinates. Typically, the condition for a subspace Wto be a point of an infinite Grassmann manifoldis that it be commensurable to a fixed subspaceH of V , in a suitable sense, either involving thedimensions of (H + W )/H and W/(H ∩ W ) orsome more analytic properties.initial object An object I in a category C withthe property that, for any object X in C thereexists a unique morphism f ∈ HomC (I, X).initial ordinalIf α is an ordinal, let |α| denote the cardinality of {τ : τ < α}. The initial ordinal corresponding to a fixed cardinal κis the minimum ordinal α such that |α| = κ.For example, ω is the initial ordinal corresponding to ℵ0 even though there are infinitely manydifferent ordinals whose cardinality is ℵ0 .
Seeordinal.infinite setAny set that is not finite. Equivalently, an infinite set is a set whose cardinalityis not a natural number. See finite set.injection (1) A one-to-one function betweentwo sets X and Y . See one-to-one function.infinite Stiefel manifoldThe infinite Stiefelmanifold Vk of k-frames is the direct limit (union)of the spaces Vn,k of k-frames in real or complex n-dimensional space.
More precisely, letF = R∞ (resp., C∞ ) denote the vector space ofinfinite sequences x = (x1 , x2 , ...) of real (resp.,complex) numbers that have only finitely manynonzero terms. Then the Stiefel manifold Vk (F )of k-frames in F is the open subset of F k consisting of k-tuples of linearly independent vectorsin F .infinityThe symbol ∞ was first used bythe English mathematician John Wallis (16161703) to denote infinity. While not a number itself, ∞ is usually used to denote a quantity thatis larger than every number. There are a number of ways mathematicians have attempted to“quantify” infinity.
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