Localization of antisymmetric spaces in the framework of quasi-metrics
Keywords:
$T_0$-quasi-metric, symmetric path, connected graph, asymmetric norm, local antisymmetric connectednessAbstract
Antisymmetric $T_0$-quasi-metric spaces which are in some sense opposite to metric spaces had appeared in the previous studies rather naturally, within the framework of asymmetry of the $T_0$-quasi-metric spaces.
In this paper, the locality status of antisymmetric $T_0$-quasimetrics is described and studied under the name local antisymmetricness. Following that we examine the cases under which conditions a non-metric $T_0$-quasi-metric space would become locally antisymmetric as well as all finite $T_0$-quasi-metric spaces are locally antisymmetric. Moreover, some asymmetric properties of locally antisymmetric $T_0$-quasi-metric spaces are determined via topological perspectives and metrics. As another approach in the context of asymmetric topology, some different aspects of the local antisymmetricness are discussed especially for the $T_0$-quasi-metrics generated by the asymmetric norms.
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