The n-dimensional sphere is the set of all points in space at a given radius from a center point. The radius is denoted by and the center by .
Unless otherwise stated, the center of the sphere is assumed to be and the radius is assumed to be 1. Under this definition, is the two points in , is the unit circle in , and is the object which is commonly called the sphere or globe, in .
An n-Disk is simply the set of all points inside , or all points less than or equal to units away from the center .
Under this definition, the 1-disk is the line segment between and on , the 2-disk is the interior of the circle of radius in , and the 3-disk or 3-ball is the interior of the sphere of radius in . As above, the center is assumed to be and the radius to be 1 unless otherwise specified.
Throughout this paper I will use the term manifold in its standard topological sense. A n-dimensional manifold is usually defined as a set with a collection of patches, or 1-1 functions from , where is an open subset of , such that the following two conditions are satisfied:
(i.) The images of the patches cover
(ii.) For two patches and , the composition and are differentiable and defined on open sets in (O'Neill).
The operator is the Cartesian product. It takes two manifolds of dimension and , and duplicates the one at every point along the other, creating a manifold of dimension .
A generator is a manifold contained in another manifold such that crossed with some other manifold makes . In other words, if , then and are generators of . For example, since , the generators of are the two .
An equator inside is a . This can be seen by setting one of the coordinates, say , to zero, and taking all remaining points in .
It's fairly trivial to see that two line segments, each end of one segment joined to an end of the other segment, make a circle (or something that can be deformed into a circle). However, the process generalizes to all dimensions, and provides us with a technique of visualizing .
Two 2-disks, or filled circles, can be joined along their circular boundaries, to make . Each disk makes a hemisphere, and the boundary between the two becomes the equator.
In general, an equator as defined above is in . Now, all the other points in are the points where . So the other points in are split into two groups: and . We can see that these two sets of points are N-disks because we can let range from 1 to 0, and get a corresponding for each value of .