Unsymmetric T₀-quasi-metrics

H.J.K. Junnila [9] called a neighbournet N on a topological space X unsymmetric provided that for each x,y∈X with y∈(N∩N⁻¹)(x) we have that N(x)=N(y). Motivated by this definition, we shall call a T₀-quasi-metric d on a set X unsymmetric provided that for each x,y,z∈X the following variant of the triangle inequality holds: d(x,z)≤d(x,y)∨d(y,x)∨d(y,z). Each T₀-ultra-quasi-metric is unsymmetric. We also note that for each unsymmetric T₀-quasi-metric d, its symmetrization d^s=d∨d⁻¹ is an ultra-metric. Furthermore we observe that unsymmetry of T₀-quasi-metrics is preserved by subspaces and suprema of nonempty finite families, but not necessarily under conjugation. In addition we show that the bicompletion of an unsymmetric T₀-quasi-metric is unsymmetric. The induced T₀-quasi-metric of an asymmetrically normed real vector space X is unsymmetric if and only if X={0}. Our results are illustrated by various examples. We also explain how our investigations relate to the theory of ordered topological spaces and questions about (pairwise) strong zero-dimensionality in bitopological spaces.