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metric space
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(Definition)
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A metric space is a set  together with a real valued function
 (called a metric, or sometimes a distance function) such that, for every
 ,
For  and
 with
 , the open ball around  of radius
 is the set
 . An open set in  is a set which equals an arbitrary (possibly empty) union of open balls in  , and  together with these open sets forms a Hausdorff topological space. The topology on  formed by these open sets is called the metric topology, and in fact the open sets form a basis for this topology (proof).
Similarly, the set
 is called a closed ball around  of radius
 . Every closed ball is a closed subset of  in the metric topology.
The prototype example of a metric space is
 itself, with the metric defined by
 . More generally, any normed vector space has an underlying metric space structure; when the vector space is finite dimensional, the resulting metric space is isomorphic to Euclidean space.
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- J.L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955.
Footnotes
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- This condition can be replaced with the weaker statement
 without affecting the definition.
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"metric space" is owned by djao.
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(view preamble)
See Also: neighborhood, vector norm, Hausdorff space, ultrametric, quasimetric space, normed vector space, pseudometric space
| Also defines: |
distance metric, metric, distance, metric topology, open ball, closed ball |
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Cross-references: Euclidean space, isomorphic, finite dimensional, vector space, structure, normed vector space, closed subset, basis, topology, Hausdorff topological space, union, open set, radius, weaker, equality, function, real
There are 343 references to this entry.
This is version 10 of metric space, born on 2001-10-25, modified 2008-02-07.
Object id is 498, canonical name is MetricSpace.
Accessed 83150 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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