Splitting metrics by T0T0-quasi-metrics

It is well known that by Szpilrajn's Theorem each partial order on a set can be extended to a linear order. In this paper we study variants of that theorem that hold for T0T0-quasi-metrics instead of partial orders.More specifically, given a metric space (X,m)(X,m), we call a T0T0-quasi-metric d defined on the set X m  -splitting provided that its symmetrization ds=d∨d−1ds=d∨d−1 is equal to m.For a given metric space (X,m)(X,m) we obtain results about T0T0-quasi-metrics on X that are minimal among the collection of all m  -splitting T0T0-quasi-metrics on X.We are also interested in the maximal possible specialization orders of m  -splitting T0T0-quasi-metrics.