## Necessary and sufficient conditions of Euclideness

*Updated November 17, 2003*

- The world function s
*(P,Q) *is symmetric

s *(P,Q)=* s *(Q ,P)*

*
*II. There exist such *n+1* points *P*^{n}º {P_{0},P_{1},...P_{n}}, that* *the* n*th order determinant* F*_{n}(P^{n})=det||*(***P**_{0}**P**_{i} .**P**_{0}**P**_{k})||, *i,k=1,2,...n *does not vanish, but *F*_{k}(P^{k})=0, for any* k+1* points *P*^{k},* *if * k>n.*

III. The world function s *(P,Q) *for arbitrary points *P,Q *can be represented in the form

*s (P,Q)=g*^{ik}(P^{n})(x_{i}(P)- x_{i}(Q))( x_{k} (P)- x_{k} (Q))/2,

where summation is produced over repeated indices from* 1 *to* n. *Here the following designations are used

*x*_{i}(P) = (**P**_{0}**P**_{i} .**P**_{0}**P**), x_{i}(Q) = (** P**_{0}**P**_{i} .**P**_{0}**Q**)

*g*_{ik}(P^{n}) = (**P**_{0}**P**_{i} .**P**_{0}**P**_{k}),

and the matrix *g*^{ik}(P^{n}) is the matrix reverse to *g*_{ik}(P^{n})

*
*g^{ik}(P^{n}) g_{lk}(P^{n})=d ^{i }_{l}

IV. Let *n* relations *x*_{i}(P) = (**P**_{0}**P**_{i} .**P**_{0}**P**) = y_{i}, where *y*_{i} , 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