Cavagnaro, Haight - Dictionary of Classical and Theoretical Mathematics (523108), страница 26

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Though the notion of an object is aprimitive in category theory, objects can be understood as generalizing or abstracting concretemathematical entities. The following categoriesare standard examples (the objects are listedfirst, the morphisms second): topological spacesand continuous functions; Abelian groups andgroup homomorphisms; rings and ring homomorphisms; partially ordered sets and monotonefunctions; complex Banach spaces and boundedlinear transformations; sets and injective functions; sets and surjective functions.obliquetal.obstruction cocycleA cocycle that represents an obstruction class in cohomology.

Seeobstruction class.obtuse angleAn angle greater than 90◦ and◦less than 180 .obtuse triangle A triangle containing an obtuse angle. See obtuse angle.octagonA polygon having eight sides.octahedronA polyhedron with eight faces.The regular octahedron, one of the five platonicsolids, has 8 triagonal faces, 12 edges, and 6vertices.omits A model A of a theory T omits a type if and only if it does not realize it. That is,A omits if and only if there is no n-tuple āof elements of A such that A |= φ(ā) for everyφ(x̄) in (x̄).Neither perpendicular nor horizon-oblique angle180◦ , or 270◦ .oblique cylindercylinder.Any angle that is not 0◦ , 90◦ ,A cylinder that is not a rightoblique triangle A triangle that does not contain a right angle.obstruction classA cohomology class orhomotopy class of maps for which being nullhomologous or homotopic to zero is equivalentto the existence of the extension of some map.Example: Suppose X is formed from a spaceA by attaching an n-cell D n to A along its boundary; that is, X is the union of A and D n with eachpoint in S n−1 , the boundary of D n , identifiedwith some point in A by a map δ.

Let f : A −→Y be a map; it will extend to a map X −→ Yexactly when the class given by f ◦ δ is zeroin the cohomology group H n (X, A; πn−1 (Y )).In particular, a map from a sphere S n−1 can beextended to a map from the disk D n exactlywhen its class in H n (D n , S n−1 ; πn−1 (Y )) =H n (S n ; πn−1 (Y )) is the zero class.© 2001 by CRC Press LLCone (1) The smallest positive integer, denoted1.(2) The multiplicative identity of the complexnumbers (and therefore of the real numbers, therational numbers, and the integers).

That is, if zis a complex number, then 1 · z = z · 1 = z.one-point compactificationA compactspace Xc obtained from a given topologicalspace X by adjoining a single point ∞ to X. Thedefinition of the topology on Xc requires that Xbe a locally compact Hausdorff space. The opensets in Xc are then defined to be the open setsof X and any set of the form V ∪ {∞} where Vis an open subset of X whose complement in Xis a compact set. Note that X is a subspace ofXc .

The one-point compactification of the realline is homeomorphic to a circle, while the onepoint compactification of the plane is homeomorphic to a sphere. The latter example is especially important in complex analysis where thehomeomorphism is called stereographic projection. See stereographic projection.one-to-one correspondenceAny functionthat is both one-to-one (injective) and onto (surjective); also known as a bijective function, or abijection. For example, the function f : R → Rone-to-one functiongiven by f (x) = 3x − 2 is a one-to-one correspondence.one-to-one function Any function f : A →B, where A and B are arbitrary sets, such thatfor every x, y ∈ A, f (x) = f (y) implies x =y. Also known as an injective function, or aninjection.

For example,the function f : N → R√given by f (n) = n is one-to-one.onto If A and B are arbitrary sets, any function f : A → B such that for every y ∈ B thereexists x ∈ A satisfying f (x) = y is an ontofunction. Also known as a surjective function,or as a surjection. For example, the functionf : R → R given by f (x) = x 3 is onto.open ballIn a metric space X, any set ofthe form B = {y : d(x, y) < r}, for somecenter x ∈ X and radius r > 0. In a metricspace, the set of open balls forms a basis forthe metric topology.

For example, in R3 withthe usual distance metric, the open balls are justthe interiors of spheres.open coverAn open cover of a subspace Aof a topological space X is a collection {Uα }of open subsets of X such that the union of allthe Uα contains A. Open covers figure in thedefinition of compactness. See compact.open n-ball with centerx = (x1 , x2 , . . . , xn ) ∈ Rnand radius r > 0 ifB = {y ∈ Rn : d(x, y) < r} ,whered(x, y) =(x1 − y1 )2 + (x2 − y2 )2 + · · · + (xn − yn )2 .open setA subset U of a topological spaceX which belongs to the topology on X.open simplexThe interior Int(σ ) of a simplex σ . Specifically, Int(σ ) = σ \ Bd(σ ), whereBd(σ ), the boundary of σ , is the union of allproper faces of σ.

For example, an open 1simplex is an open interval, while an open 2simplex is the interior of a triangle. See simplex.open starIf S is a simplicial complex andv is a vertex of S, the open star of the vertex vis defined to be the union of the interiors of allsimplices σ of S that have v as a vertex. Seesimplicial complex.open disk An open ball in R2 with the usualdistance metric. See open ball. That is, D is anopen disk with center x = (x1 , x2 ) ∈ R2 andradius r > 0 ifopposite anglesTwo angles on a polygon(having an even number of sides) having an equalnumber of angles between them, regardless ofthe direction around which one counts.D = {y ∈ R2 : d(x, y) < r}= {(y1 , y2 ) : (x1 − y1 )2 + (x2 − y2 )2 < r} .opposite angle/side A side and an angle on apolygon (with an odd number of sides) having anequal number of sides between them, regardlessof the direction around which one counts.open formula A well-formed formula α of afirst order language L such that α is quantifierfree; i.e., α does not have any quantifiers.opposite sidesA pair of sides on a polygon (having an even number of sides) having anequal number of sides between them, regardlessof the direction around which one counts.open mapA function f : X → Y such thatthe image f (U ) of any open set U of X is anopen set in Y .

If f is invertible, then f is an openmap if and only if f −1 : Y → X is continuous.open n-ball An open ball in Rn with the usualdistance metric. See open ball. That is, B is an© 2001 by CRC Press LLCordered n-tuple A list of n arbitrary objectswith a specified order, viewed as a single object.The first component of the n-tuple is the objectlisted first, the nth component is the√object listedlast, etc. For example, (10, 10, π, 2, b) is anordered 5-tuple; the third component is π .orientationordered pair An ordered list of two objects.The first (second) component of the ordered pairis the object listed first (second). For example,(−1, A) is an ordered pair with −1 and A as itsfirst and second components, respectively.ordered setSee partially ordered set.ordered triple An ordered n-tuple with n =3.

See ordered n-tuple.ordering A partial ordering (on a set A). Seepartial ordering.order topology The topology on a set X, witha linear order relation, with a basis consisting ofall intervals of the form (a, b) for any a, b ∈X. If X has either a minimal element m or amaximal element M, then the sets [m, b) and(a, M] are included as well. On the real line, theorder topology is the standard topology; that is,the topology with a basis consisting of the openintervals.order type (of a well-ordered set) The uniqueordinal number that is order-isomorphic to thegiven well-ordered set.

Thus, the set {−2, 1, 5},which is well ordered by the relation −2 < 1 <5, has order type 3. The set N ∪ {#}, which iswell ordered by the relation 0 < 1 < 2 < 3 <· · · < #, has order type ω + 1.ordinal (or ordinal number)A transitiveset that is strictly well ordered by the elementrelation ∈.

For example, the ordinal number 3is the set {0, 1, 2}; it is a transitive set and it iswell ordered by ∈.ordinary helixA curve lying on a cylinderwhich forms a constant angle with the elementsof the cylinder.ordinateThe y-coordinate of a point in theCartesian xy-plane is the ordinate of that point.For example, the ordinate of the point (2, −3)is −3.orientable fiber bundleA fiber bundleF −→ E −→ B, with F a connected compactn-manifold, such that it is possible to chooseelements in the homology Hn (Fb ) of the fiber© 2001 by CRC Press LLCabove each point b in B so that around eachpoint there is a neighborhood U and a generator of the homology Hn (E|U ) of E restricted toU so that the inclusion of the fiber into E|U induces a map that takes the (chosen) generator ofHn (Fb ) to the (chosen) generator of Hn (E|U ).Examples: Any trivial bundle over an orientable manifold is again orientable.

But theMöbius band, as a bundle over S 1 , is not orientable.orientation A specific choice of direction fora vector space, simplex, or cell. The definitionsof orientation for these objects extend to give theimportant notions of an orientation on a manifold, simplicial complex, or cell complex.(1) For an n-dimensional vector space V ,an orientation is determined by the choice ofan ordered basis {v1 , .

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