Metrizability of asymmetric spaces

Recent research has spawned an evolving application of non-Hausdorff topologies [22] generated by nonsymmetric distance functions to areas within mathematics (e.g., posets and continuous lattices [6]) as well as to allied areas that might seem remote (e.g., computer science ([24],[4],[16]) and biology [25]). The study of nonsymmetric distance can be traced to the 1910 thesis of T.H. Hildebrandt [11], written at the University of Chicago under the direction of E.H. Moore. In a spirit parallelling that of Hildebrandt's, this paper undertakes a study of nonsymmetric distance for which an equivalent symmetric distance can be constructed. This illuminates situations in which an associated metric exists. Of particular interest in our study are distances that are either locally symmetric or locally satisfy the triangle inequality. We recall Niemytzki's classical result [19] that a topological space is metrizable if, and only if, there is a semimetric for the space that locally satisfies the triangle inequality. We investigate two forms in which a nonsymmetric distance might locally satisfy the triangle inequality. In either case, we show that such spaces are metrizable, when the distance is locally symmetric. Although many of our results are known, the approach is particularly straightforward in providing an explicit construction of a distance with the desired properties.