The definitions of open balls, closed balls and spheres within a metric space are introduced.. Ball (mathematics) In Euclidean space, a ball is the volume bounded by a sphere. In mathematics, a ball is the solid figure bounded by a sphere; it is also called a solid sphere. [1] It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them). These concepts are defined not only in three.
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Considering only open or closed balls will not be general enough for our domains. To generalize open and closed intervals, we will consider their boundaries and interiors. By our definition, the boundary of an interval is the set of two endpoints. Then we categorize types of intervals by whether they contain all of their boundary points or not.. After introducing open and closed balls, we showed that all open sets are unions of open balls and that boundary, closure and interior can be identified using open balls. We showed that balls in normed linear spaces are all convex and balanced and that, in any given space, they all have the same shape. Download to read the full chapter text.
