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.

中文翻译：

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非对称 T0 准度量

摘要 HJK Junnila [9] 称拓扑空间 X 上的邻居网络 N 不对称，前提是对于每个 x , y ∈ X 且 y ∈ ( N ∩ N − 1 ) ( x ) 我们有 N ( x ) = N ( y ) . 受此定义的启发，我们将称集合 X 上的 T 0 -拟度量 d 为非对称的，前提是对于每个 x , y , z ∈ X 三角不等式的以下变体成立： d ( x , z ) ≤ d ( x , y ) ∨ d ( y , x ) ∨ d ( y , z ) 。每个T 0 -超准度量是不对称的。我们还注意到，对于每个非对称 T 0 -拟度量 d，其对称化 ds = d ∨ d − 1 是一个超度量。此外，我们观察到 T 0 -拟度量的不对称性由非空有限族的子空间和上位保留，但不一定在共轭下。此外，我们表明非对称 T 0 -拟度量的双完成是非对称的。非对称赋范实向量空间 X 的诱导 T 0 -拟度量是非对称的当且仅当 X = { 0 } 。我们的结果用各种例子来说明。我们还解释了我们的研究如何与有序拓扑空间理论和关于位拓扑空间中（成对）强零维的问题相关。