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quasimetric space
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(Definition)
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A quasimetric space  is a set  together with a non-negative real-valued function
 (called a quasimetric) such that, for every
 ,
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:
- If
 is a quasimetric space, we can form a metric space  where  is defined for all  by
- Every metric space is trivially a quasimetric space.
- A quasimetric that is symmetric (i.e. satisfies
 for all  is a metric.
- 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.
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"quasimetric space" is owned by mathcam. [ full author list (2) ]
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Cross-references: metric, properties, axiom, symmetry, points, metric space, equality, function
There are 3 references to this entry.
This is version 5 of quasimetric space, born on 2004-10-02, modified 2006-06-17.
Object id is 6274, canonical name is QuasimetricSpace.
Accessed 4457 times total.
Classification:
| AMS MSC: | 54E35 (General topology :: Spaces with richer structures :: Metric spaces, metrizability) |
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Pending Errata and Addenda
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