DEFINITING SHAPES as SETS of VECtors?

Question

Could I define shapes as sets of vectors like

A sphere:

Let V = <x,y,z>

then S = { V : |V|=sqrt(r) }

Is a sphere of radius r

and since vectors have no position ...I could say

Two figures G1 and G2 are topologically the same iff

G1=S AND G2=S .

Answer 1

your thoughts are valid but your example of a sphere via S is not...can you see why not??...try a sphere of radius 4...and how would you describe the unit cube via a P(V)...P(V) would be fairly complicated