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Cone of a circle. The original space X is in blue, and the collapsed end point v is in green.. In topology, especially algebraic topology, the cone of a topological space is intuitively obtained by stretching X into a cylinder and then collapsing one of its end faces to a point.
According to the above definition, if C is a convex cone, then C ∪ {0} is a convex cone, too. A convex cone is said to be pointed if 0 is in C, and blunt if 0 is not in C. [2] [21] Blunt cones can be excluded from the definition of convex cone by substituting "non-negative" for "positive" in the condition of α, β.
A cone from N to F is a family of morphisms : for each object X of J, such that for every morphism f : X → Y in J the following diagram commutes: Part of a cone from N to F. The (usually infinite) collection of all these triangles can be (partially) depicted in the shape of a cone with the apex N.
A topological space X is locally contractible at a point x if for every neighborhood U of x there is a neighborhood V of x contained in U such that the inclusion of V is nulhomotopic in U. A space is locally contractible if it is locally contractible at every point. This definition is occasionally referred to as the "geometric topologist's ...
The definition of a cone may be extended to higher dimensions; see convex cone. In this case, one says that a convex set C in the real vector space R n {\displaystyle \mathbb {R} ^{n}} is a cone (with apex at the origin) if for every vector x in C and every nonnegative real number a , the vector ax is in C . [ 2 ]
Let X be a projective space over some field K, and R, S be disjoint subspaces of X. Let A be an arbitrary subset of S. Then we define RA, the cone with top R and basis A, as follows : When A is empty, RA = A. When A is not empty, RA consists of all those points on a line connecting a point on R and a point on A.
Let be a proper variety. By definition, a (real) 1-cycle on is a formal linear combination = of irreducible, reduced and proper curves , with coefficients . Numerical equivalence of 1-cycles is defined by intersections: two 1-cycles and ′ are numerically equivalent if = ′ for every Cartier divisor on .
If the cone C=Spec X R is the total space of a vector bundle E, then O(-1) is the tautological line bundle on the projective bundle P(E). Remark : When the (local) generators of R have degree other than one, the construction of O (1) still goes through but with a weighted projective space in place of a projective space; so the resulting O (1 ...