a vector of norm 1
a ball of radius r
an element of finite order
the group of invertible elements of E
Actually, S has the much stronger property of being convex.
This method has the disadvantage of not being intrinsic.
However, only five of these are distinct.
The best known of these is the Knaster continuum.
Of these, (i) and (ii) are almost immediate from the definition.
Suppose that of all such solutions, (x,y,z) is one with y minimal.
Theorem 2 of 
statement (ii) of Proposition 7
However, this cannot be proved of the cardinal function d(X).
For example, F reaches a relative maximum of 5.2 at about x=2.1.