quasimetric space
A quasimetric space is a set together with a non-negative real-valued function (called a quasimetric) such that, for every ,
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with equality if and only if .
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In other words, a quasimetric space is a generalization of a metric space in which we drop the requirement that, for two points and , the “distance” between and is the same as the “distance” between and (i.e. the symmetry axiom of metric spaces).
Some properties:
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If is a quasimetric space, we can form a metric space where is defined for all by
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Every metric space is trivially a quasimetric space.
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A quasimetric that is (i.e. for all is a metric.
References
- 1 L.A. Steen, J.A.Seebach, Jr., Counterexamples in topology, Holt, Rinehart and Winston, Inc., 1970.
- 2 Z. Shen, Lectures of Finsler geometry, World Sientific, 2001.
Title | quasimetric space |
Canonical name | QuasimetricSpace |
Date of creation | 2013-03-22 14:40:21 |
Last modified on | 2013-03-22 14:40:21 |
Owner | mathcam (2727) |
Last modified by | mathcam (2727) |
Numerical id | 8 |
Author | mathcam (2727) |
Entry type | Definition |
Classification | msc 54E35 |
Synonym | quasi-metric space |
Related topic | PseudometricSpace |
Related topic | MetricSpace |
Related topic | GeneralizationOfAPseudometric |
Defines | quasimetric |
Defines | quasi-metric |