Noun
n-sphere (plural n-spheres)
An n-dimensional hypersphere: the 2-sphere is a circle, the 3-sphere is the usual sphere, the 4-sphere is a glome, etc.
A stronger assumption is necessary; in dimensions four and higher there are simply-connected, closed manifolds which are not homotopy equivalent to an n-sphere. Source: Internet
For dimensions greater than three, one can pose the Generalized Poincaré conjecture: is a homotopy n-sphere homeomorphic to the n-sphere? Source: Internet
Additionally, each result can be reduced to the other result in its column. citation Generalizations 1. In the original BUT, the domain of the function f is the unit n-sphere (the boundary of the unit n-ball). Source: Internet
An (n + 1)-ball is closed if it includes the n-sphere, and it is open if it does not include the n-sphere. Source: Internet
BUT is equivalent to the following statement: A continuous odd function from an n-sphere into Euclidean n-space has a zero. Source: Internet
Interestingly, given the radius R, the volume and the surface area of the n-sphere reaches a maximum and then decrease towards zero as the dimension n increases. Source: Internet