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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 ...
The lower half-plane is the set of points (,) with < instead. Arbitrary oriented half-planes can be obtained via a planar rotation . Half-planes are an example of two-dimensional half-space .
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.
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 ...
Rigor is a cornerstone quality of mathematics, and can play an important role in preventing mathematics from degenerating into fallacies. well-behaved An object is well-behaved (in contrast with being Pathological ) if it satisfies certain prevailing regularity properties, or if it conforms to mathematical intuition (even though intuition can ...
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.
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 ...
In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. [1] [2] [3] That is, a function : is open if for any open set in , the image is open in . Likewise, a closed map is a function that maps closed sets to closed sets.