Cavagnaro, Haight - Dictionary of Classical and Theoretical Mathematics (523108), страница 38
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That is, the mapsg → g −1 from G to G and (g1 , g2 ) → g1 g2from G × G to G are continuous.Any discrete group is considered to be a topological group with the discrete topology thatstates: any single element subset is an open set.topological invariantA property preservedby homeomorphisms. That is, P is a topological invariant if, given any homeomorphism f :X → Y , the space X has property P if andonly if Y has property P .
For example, connectedness, separability, and normality are alltopological invariants.total orderingtotient functionSee linear ordering.See Euler phi function.transcendental numberber.See algebraic num-transfinite cardinalAny infinite cardinalnumber. For example, ℵ3 is a transfinite cardinal.truth tabletransfinite inductionA method of proof.Suppose P (α) is some statement that describesa property of α, where α is an ordinal. Supposethat all of the following conditions hold: (i.)P (α0 ), for some α0 , (ii.) P (α) implies P (α+1),for all α ≥ α0 , and (iii.) (∀β < λ) P (β) impliesP (λ), for any nonzero limit ordinal λ.
Fromthese three, the conclusion is that P (α) holdsfor all ordinals α ≥ α0 . Transfinite induction isa generalization of induction.transfinite ordinalAny ordinal that is infinite. For example, ω + 3 is a transfinite ordinal.transfinite recursionA method of defining some function; also known as definition bytransfinite recursion, or sometimes as definitionby transfinite induction.
For any function gon the universe of sets, there exists a uniquefunction f on the class of ordinals such thatf (α) = g(f |α), for all ordinals α. See alsorecursion.transitive relation A binary relation R suchthat [(x, y) ∈ R] ∧ [(y, z) ∈ R] implies (x, z) ∈R, for all x, y, z. For example, ≤ is a transitiverelation on N since if n ≤ m and m ≤ k, thenn ≤ k.transitive setA set A such that, wheneverB ∈ A, then B ⊆ A.tree A partial order (T , ≤) in which, for anyt ∈ T , the set of predecessors of t, {s ∈ T :s < t}, is well ordered by <. That is, any nonempty subset of {s ∈ T : s < t} has a leastelement. An example of a tree is the set of allfinite sequences of natural numbers, ordered byextension: s < t if t extends s.
Other examples include Aronszajn trees, Kurepa trees, andSuslin trees. See Aronszajn tree, Kurepa tree,Suslin tree.triangular numberThe integers in the sequence 1, 3, 6, 10, . . . (which represent the number of lattice points in the plane that lie on theperimeter of isosceles right triangles having integer length legs).Thetriangular numbers are integers of theform nk=1 k.© 2001 by CRC Press LLCtruth assignmentIn propositional logic, afunction v : S → {T , F } mapping a set S ofsentence symbols to {T , F }, where T is interpreted as true and F is interpreted as false. Forexample, if S = {A1 , A2 , A3 }, then a possibletruth assignment would be v : S → {T , F } byv(A1 ) = F , v(A2 ) = T , and v(A3 ) = T .
Notethat there are eight possible truth assignmentsfor this particular set of sentence symbols, sincethere are two choices (T or F ) for each valueof the function on an element of S. In general,if S has n sentence symbols, then there are 2npossible truth assignments on S.A truth assignment v : S → {T , F } is extended using a recursive definition to a truthassignment v on the set S of all well-formedpropositional formulas α which have sentencesymbols from S, as follows:(i.) If α is a sentence symbol in S, thenv(α) = v(α).(ii.) If α = (¬β), thenT if v(β) = Fv(α) =F if v(β) = T .(iii.) If α = (β ∧ γ ), thenTif v(β) = v(γ ) = Tv(α) =Fotherwise.(iv.) If α = (β ∨ γ ), thenTif v(β) = T or v(γ ) = Tv(α) =Fotherwise.(v.) If α = (β → γ ), thenTif v(β) = F or v(γ ) = Tv(α) =Fotherwise.(vi.) If α = (β ↔ γ ), thenT if v(β) = v(γ )v(α) =F if v(β) = v(γ ).truth table A table of truth values for a wellformed propositional formula α, based on assignments of truth values for the sentence symbols in α.
In general, if there are n sentencesymbols in α, then the truth table will have 2nrows. The truth tables for the formulas built uptubular neighborhoodfrom the logical connectives (here A and B arewell-formed propositional formulas) are as follows, where T is interpreted as true and F isinterpreted as false.ATF(¬A)FTATTFFBTFTF(A ∧ B)TFFFATTFFBTFTF(A ∨ B)TTTFATTFFBTFTF(A → B)TFTTATTFFBTFTF(A ↔ B)TFFTA truth table for the more complicated wellformed propositional formula ((A ∨ B) → C),where A, B, C are sentence symbols, is as follows.ATTTTFFFFBTTFFTTFFCTFTFTFTF(A ∨ B)TTTTTTFF((A ∨ B) → C)TFTFTFTTtubular neighborhood A tubular neighborhood of a simple closed curve L ⊂ S 3 is a neighborhood of L homeomorphic to L × B 2 whereL × {0} is identified with the curve L.© 2001 by CRC Press LLCMore generally, a tubular neighborhood ofan l-dimensional submanifold L ⊂ M in an ndimensional manifold M is a neighborhood ofL homeomorphic to L × B m−l .Turing complete set A set A of natural numbers which is recursively enumerable and, forany recursively enumerable set B, B ≤T A;i.e., B is computable, relative to A.An example of a Turing complete set is thehalting set K = {e : ϕe (e) is defined}, where ϕedenotes the partial recursive function with Gödelnumber e.Turing complete is sometimes simply referredto as complete.Turing equivalent Two sets A and B of natural numbers such that A is Turing reducible toB (A ≤T B) and B is Turing reducible to A(B ≤T A).
Intuitively, Turing equivalent setsare sets that code the same information. Turingequivalence (notation: A ≡T B) is an equivalence relation on the class of all sets of naturalnumbers. The equivalence classes of ≡T arecalled Turing degrees, or degrees of unsolvability.As an example, any two Turing complete setsare Turing equivalent.Turing reducibilityLet ϕ be a partial function on N; i.e., its domain is some subset of N,and let A be a set of natural numbers. The function ϕ is Turing reducible to A if ϕ is (Turing)computable, relative to A.
See relative computability. The notation ϕ ≤T A means that ϕis Turing reducible to A. If B is a set of natural numbers, then B is Turing reducible to A(B ≤T A) if its characteristic function χB isTuring reducible to A.For example, given any set A of natural numbers, A ≤T A where A is the complement of Ain N. If B is any computably enumerable (recursively enumerable) set and K is the haltingset {e : ϕe (e) is defined}, where ϕe is the partialrecursive function with Gödel number e, thenB ≤T K.twin primes Two odd prime numbers p andq so that q = p + 2. For example, 3 and 5 aretwin primes, as are 5 and 7, 11 and 13, 17 and19, and 29 and 31. Twin primes with over 3300typedigits have been discovered, but it is unknownwhether or not there are infinitely twin primepairs. The triple (3, 5, 7) forms the only “primetriplet” since at least one of any triple of the form(n, n + 2, n + 4) must be divisible by 3.Tychonoff Fixed-Point TheoremSupposeX is a locally convex linear topological spaceand C ⊆ X is compact and convex.
Then anycontinuous function f : C → C has a fixedpoint. That is, there is a c ∈ C with f (c) = c.Any normed vector space can be made into alocally convex linear topological space by using the metric topology generated by the norm:d(x, y) = x − y.Tychonoff space See completely regular topological space.Tychonoff TheoremThe product ofany number of compact topological spaces is© 2001 by CRC Press LLCcompact in the product topology. For example,since the unit interval [0, 1] is compact, any cube[0, 1]κ is also compact. It is this theorem thatmakes the product (Tychonoff) topology important.
The Tychonoff Theorem is equivalent tothe Axiom of Choice.Tychonoff topologySee product topology.type A type of a theory T is any set of formulas that is realized in some model of T . That is,if T is a (possibly empty) theory in the languageL, then a set of L-formulas (x̄) is an n-typeof T if x̄ = {x1 , . . . , xn } contains all free variables occurring in the formulas of , and thereis a model A of T and an n-tuple ā of elements ofA such that A |= φ(ā) for every φ(x̄) in (x̄).Some authors require types to be complete,meaning they are maximally consistent.uniform spaceUultrafilterA subset U of a Boolean algebraB, which is a filter, not properly contained inany other filter on B.
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