Updated November 17, 2003
s
(P,Q)= s (Q ,P)II. There exist such n+1 points Pnº {P0,P1,...Pn}, that the nth order determinant Fn(Pn)=det||(P0Pi .P0Pk)||, i,k=1,2,...n does not vanish, but Fk(Pk)=0, for any k+1 points Pk, if k>n.
III. The world function s (P,Q) for arbitrary points P,Q can be represented in the form
s
(P,Q)=gik(Pn)(xi(P)- xi(Q))( xk (P)- xk (Q))/2,where summation is produced over repeated indices from 1 to n. Here the following designations are used
xi(P) = (P
0Pi .P0P), xi(Q) = ( P0Pi .P0Q)and the matrix gik(Pn) is the matrix reverse to gik(Pn)
gik(Pn) glk(Pn)=d i l
IV. Let n relations xi(P) = (P0Pi .P0P) = yi, where yi , i=1,2,...n, are arbitrary real numbers, be considered to be equations for determnation of the point P. Then these equations have always one and only one solution.
All four conditions are formulated in terms of the world function and only in terms of the world fuction. The condition II determines the dimension of the space. The condition III introduces a rectilinear coordinate system in the space, determines a linear space in the space and gives the scalar product. The condition IV describes topological properties of the space (in particular continuity of the space)
Updated November 17, 2003