Cavagnaro, Haight - Dictionary of Classical and Theoretical Mathematics (523108), страница 24
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An element x ∈ P is a lowerbound for S if x ≤ s for all s ∈ S.lower limit topologySee Sorgenfrey line.Luzin spaceAn uncountable regular topological space that has no isolated points and inwhich every nowhere dense set is countable. ALuzin space is hereditarily ccc and hereditarily Lindelöf. (See countable chain condition.)Because of this, Luzin spaces are related to Lspaces, which are hereditarily Lindelöf but nothereditarily separable.mean curvatureMmagic square A square array of positive integers such that the sum of all its rows, columns,and diagonals are equal. Often an additionalcondition is added; namely, that the entries ofthe n × n magic square include all of the integers 1, 2, . . . , n2 . An example of a 3 × 3 magicsquare of this type is83415967 .2Mangoldt function The arithmetic function,denoted , which is defined as follows: (n) =log p if n = p i for some prime number pand positive integer i, and (n) = 0 otherwise.
(See arithmetic function.) For example,(8) = log 2, (15) = 0. This function playsan important part in elementary proofs of theprime number theorem.manifoldA topological space M with theproperty that each point P possesses a neighborhood that is homeomorphic to Euclidean spaceRn , for some n. If M is connected, the dimension n is constant, and M is an n-manifold. Usually, but not always, it is desirable to assume alsothat M is Hausdorff and metrizable.mappingSee function.mapping cylinder Given a map f : X → Ybetween topological spaces, the mapping cylinder If of f is the quotient space of the disjointunion of X×[0, 1] and Y obtained by identifyingeach point (x, 0) ∈ X × 0 with f (x) ∈ Y.
Thespace If is homotopy equivalent to Y and themap f : X → Y is homotopically equivalent tothe natural inclusion i : X → If . The mappingcylinder thus justifies the statement that, in homotopy theory, “every map is equivalent to aninclusion.” There is also an algebraic version ofthe mapping cylinder when X and Y are chaincomplexes.© 2001 by CRC Press LLCMartin’s Axiom (MA) If P is a partial orderwith the countable chain condition and D is acollection of fewer than continuum-many densesubsets of P, there is a filter G ⊆ P which meetsevery D ∈ D. That is, as long as P has no uncountable collections of incompatible elements(antichains), generic filters are able to meet anyset D of dense subsets with |D| < 2ω .
In a partial order, D ⊆ P is dense if for any p ∈ P thereis a q ∈ D with q ≤ p. A topological equivalentis: if D is a collection of fewer than continuummany dense open sets of a compact Hausdorffspace X with the countable chain condition, then∩D is dense in X.For an infinite cardinal κ, MAκ is the statement that if D is a collection of dense subsets ofa ccc partial order P with |D| ≤ κ, then thereis a filter G ⊆ P such that G ∩ D = ∅ foreach D ∈ D. Thus, Martin’s Axiom (MA) is theassertion that for all κ < 2ω , MAκ .MAω is a theorem of ZFC, and so the Continuum Hypothesis implies MA. However, MA isalso consistent with ¬CH.
Some consequencesof MAω1 are Suslin’s Hypothesis (there are noSuslin lines or trees), the union of ω1 measurezero sets has measure zero, and the union of ω1meager sets is meager.mathematical inductionSee induction.maximal elementGiven a set A and an ordering ≤ on A, m ∈ A is said to be a maximalelement of A if there does not exist x ∈ A withm < x. Alternatively, m ∈ A is a maximal element of A if, for all x ∈ A, if m ≤ x, thenm = x. Note that if A has a greatest or maximum element, then it is unique, and it is also theunique maximal element of A. If A has no greatest element, then A may have more than one, orno, maximal elements.
See greatest element.maximum elementA greatest element of aset A with an ordering ≤. See greatest element.mean curvatureThe arithmetical mean ofthe principal curvatures of the surface S at P . Itis half the trace of the second fundamental formof the surface at P . (Some writers do not divideby 2.) More generally, the mean curvature at Pin a hypersurface S of Rn+1 is n1 times the traceof the second fundamental form.mediantmediantThe mediant of two rational numbers pq and rs is the rational number p+rq+s . For7example, the mediant of 23 and 58 is 2+53+8 = 11 .The mediant of two positive rational numbers isalways between the two rational numbers.member of a setAny object that belongs toa given set, that is, is an element of that set. Forexample, the number 5 is a member of the set{a, 5, 2}.
Notation: x ∈ S (x is a member of S),and x ∈/ S (x is not a member of S).meridian of a sphereAn inclusion of S n−1nnin S which splits S into two equal size halves.The equator is a meridian of the sphere S 2 .Mersenne numberA number of the formMn = 2n − 1, where n is a positive integer. Determining which Mersenne numbers are primehas long interested mathematicians. See alsoMersenne prime.Mersenne prime A number of the form Mn= 2n − 1, where n is a positive integer, whichis prime. For example, M2 = 3 and M5 =31 are Mersenne primes.
If Mn is a Mersenneprime, then n is prime. However, the converseis not true: M11 = 23 · 89. In fact, Mersenneprimes are rare. There are currently 35 knownMersenne primes, the largest being M1398269 . Itis unknown whether there are infinitely manyMersenne primes.Mersenne primes are named for Marin Mersenne, a 17th century monk who made a conjecture regarding which primes p ≤ 257 aresuch that Mp is a Mersenne prime. He was latershown to have made errors of both commissionand omission in his conjecture.metric A function d : X ×X → R satisfying(i,) d(x, y) > 0 if x = y; d(x, x) = 0,(ii.) d(x, y) = d(y, x) and(iii.) d(x, y) + d(y, z) ≤ d(x, z), for allx, y, z ∈ X.A metric may be interpreted as a distancefunction on the set X.metric space A topological space X equippedwith a metric d such that the topology of X is thatinduced by d. See metric. Specifically, givenx ∈ X define the ε-ball about x by Bε (x) = {y :© 2001 by CRC Press LLCd(x, y) < ε}.
Then the ε-balls Bε (x) for allx ∈ X and ε > 0 form a basis for the topologyof X.metrizable space A topological space X suchthat there exists a metric d on X for which thetopology on X is the metric topology inducedby d. See metric, metric space.Meyer-Vietoris sequenceA long exact sequence in homology (or cohomology) that is obtained when a topological space X is the unionof two subspaces X1 and X2 such that the inclusion (X1 , X1 ∩ X2 ) → (X, X2 ) (viewed asa map of pairs) induces an isomorphism in relative homology.
The exact sequence is of theform· · · → Hp (X1 ∩ X2 ) → Hp (X1 ) ⊕ Hp (X2 ) →Hp (X) → Hp−1 (X1 ∩ X2 ) → · · ·The Mayer-Vietoris sequence is closely relatedto the Excision Theorem for singular theory.microbundleA pair of maps i : B −→ Eand j : E −→ B such that j i is the identitymap on B and for each b in B, there are openneighborhoods U of b and V of ib with iU ⊂ Vand j V ⊂ U and a homeomorphism h : V −→U × R n with the following properties.(i.) The map hi restricted to U includes U asU × {0} in U × Rn .(ii.) The map h followed by projection ontoU is equal to the restriction of j to the set V .The integer n is called the fiber dimension ofthe microbundle.Microbundles were introduced in an attemptto construct tangent bundles on manifolds without differentiable structures.
J. Milnor (Microbundles I. Topology 3 (1964) suppl. 1, 53–80)uses microbundles to show that there is a topological manifold M such that no Cartesian product M × M has a differentiable structure thatagrees with the original topological structure.minimal elementGiven a set A and an ordering ≤ on A, m ∈ A is said to be a minimalelement of A if there does not exist x ∈ A withx < m. Alternatively, m ∈ A is a minimal element of A if, for all x ∈ A, if x ≤ m, thenm = x. Note that if A has a least or minimum element, then it is unique, and it is also themodel completeunique minimal element of A.
If A has no leastelement, then A may have more than one, or no,minimal elements. See least element.minimal surface A surface in R3 with meancurvature vanishing at every point. (See meancurvature.) Equivalently, a minimal surface isa critical point for the surface area functional.This definition generalizes to surfaces in higherdimensional spaces or more general Riemannianmanifolds. A subtlety of the term is the fact thata minimal surface need not minimize area; suchsurfaces are called stable minimal surfaces.minimum elementA least element of a setA with an ordering ≤. See least element.Möbius functionThe arithmetic function,denoted µ, which is defined as follows: µ(1) =1; µ(n) = (−1)k if n is square-free and has kdistinct prime divisors; and µ(n) = 0 if n isnot square-free.
(See arithmetic function.) Forexample, µ(30) = (−1)3 = −1, µ(18) = 0. Itis multiplicative.Möbius inversion formula Let f be an arithmetic function. Define the arithmetic functionF bynF (n) =f (d) · g,ddwhere d ranges over the positive divisors of n.ThennF (d) · µ,f (n) =ddmixed areaA useful concept in convex geometry, based on the observation that one canform a weighted average of convex figures toobtain a new convex figure. If M and M1 areconvex figures in the plane and 0 ≤ s ≤ 1, themixed figure Ms is formed by taking all pointssP + tQ for which P is a point in M, Q isa point in M1 , and t = 1 − s. If A(Ms ) isthe area of the convex figure Ms , then A(Ms ) =s 2 A(M) + 2stA(M, M1 ) + t 2 A(M1 ), where thenumber A(M, M1 ) is the mixed area of M andM1 .Möbius bandThe rectangle {(x, y) ∈ R ×R : 0 ≤ x, y ≤ 1}, with the identification(0, y) ∼ (1, 1 − y) for 0 ≤ y ≤ 1.
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