Local Symmetry and Triangle Laws Are Sufficient for Metrisability

There is a long history in which attempts have been made to weaken the conditions defining a metric function whilst retaining metrisability. In this note, we survey some of our recent results in this context, deduce some well-known metrisation theorems, and present some related unsolved problems. Some of the more important early papers on this topic were by E. W. Chi t ten den (see, for example [2]). But perhaps,the most interesting of early theorems is the following. Theorem 1 (V. W. Ni em y t z k i [10]). In order that a Hausdorff space X is metrisable it is necessary and sufficient that there exist a real-valued function d on X X X generating the topology on X and satisfy ing (l') d(x,y) ~ 0, d(x,y) = 0 ifand only if :x= y" (2) d(x, y) = d(y, x), (3) given x E X, € > 0, there is 0> 0 such that d(x, z) < €-177