Search results
Results from the WOW.Com Content Network
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.
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 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]
X is homotopy equivalent to a one-point space. X deformation retracts onto a point. (However, there exist contractible spaces which do not strongly deformation retract to a point.) For any path-connected space Y, any two maps f,g: X → Y are homotopic. For any nonempty space Y, any map f: Y → X is null-homotopic. The cone on a space X is ...
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.
So the cone theorem shows that the cone of curves of a Fano variety is generated by rational curves. If in addition the variety X {\displaystyle X} is defined over a field of characteristic 0, we have the following assertion, sometimes referred to as the Contraction Theorem :
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.
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.
A cone C in a vector space X is said to be self-dual if X can be equipped with an inner product ⋅,⋅ such that the internal dual cone relative to this inner product is equal to C. [3] Those authors who define the dual cone as the internal dual cone in a real Hilbert space usually say that a cone is self-dual if it is equal to its internal dual.