LAUSR

Abstract 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 T 0 -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 T 0 -ultra-quasi-metric is unsymmetric. We also note that for each unsymmetric T 0 -quasi-metric d, its symmetrization d s = d ∨ d − 1 is an ultra-metric. Furthermore we observe that unsymmetry of T 0 -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 T 0 -quasi-metric is unsymmetric. The induced T 0 -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.