On a distance symmetry in geometry (Q1897919)

The author defines in an arbitrary set \({\mathcal M} = \{i,j,k, \dots\}\) a distance between points of \({\mathcal M}\) with the use of a functional correspondence \({\mathcal L} : {\mathcal M} \times {\mathcal M} \to \mathbb{R}\). This distance satisfies the symmetric axiom, when \({\mathcal L} (i,j) = \theta ({\mathcal L} (j,i))\), where \(\theta\) is a strictly monotoneous function of one variable whose domain and range coincide with the range of values \({\mathcal L} ({\mathcal S}_{\mathcal L})\) of the given functional correspondence \({\mathcal L}\). The following theorem is proved: if a distance between points of a space \({\mathcal M}\) satisfies the symmetry axiom, then this distance is either symmetric or, to within the equivalence, antisymmetric.