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A half-space can be either open or closed. An open half-space is either of the two open sets produced by the subtraction of a hyperplane from the affine space. A closed half-space is the union of an open half-space and the hyperplane that defines it. The open (closed) upper half-space is the half-space of all (x 1, x 2, ..., x n) such that x n ...
In mathematics, and particularly general topology, the half-disk topology is an example of a topology given to the set , given by all points (,) in the plane such that . [1] The set X {\displaystyle X} can be termed the closed upper half plane.
The uniformization theorem for surfaces states that the upper half-plane is the universal covering space of surfaces with constant negative Gaussian curvature. The closed upper half-plane is the union of the upper half-plane and the real axis. It is the closure of the upper half-plane.
A closed ball of radius r is a closed r-ball. Every closed ball is a closed set in the topology induced on M by d. Note that the closed ball D(x; r) might not be equal to the closure of the open ball B(x; r). Closed set A set is closed if its complement is a member of the topology. Closed function
Conversely, if is a closed set with nonempty interior such that every point on the boundary has a supporting hyperplane, then is a convex set, and is the intersection of all its supporting closed half-spaces. [2] The hyperplane in the theorem may not be unique, as noticed in the second picture on the right.
If α > 0, it is a closed disk of radius 1/α; If α < 0, it is the closure of the complement of a disk of radius −1/ α . Then an edge of the alpha-shape is drawn between two members of the finite point set whenever there exists a generalized disk of radius 1/ α that has the two points on its boundary and that contains none of the point set ...
In mathematics, a plane is a two-dimensional space or flat surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. When working exclusively in two-dimensional Euclidean space, the definite article is used, so the Euclidean plane refers to the ...
For example, the entire complex plane is a domain, as is the open unit disk, the open upper half-plane, and so forth. Often, a complex domain serves as the domain of definition for a holomorphic function. In the study of several complex variables, the definition of a domain is extended to include any connected open subset of C n.