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metric space
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(Definition)
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A metric space is a set $X$ together with a real valued function $d: X \times X \longrightarrow \mathbb{R}$ (called a metric, or sometimes a distance function) such that, for every $x,y,z \in X$ ,
- $d(x,y) \geq 0$ , with equality 1 if and only if $x=y$
- $d(x,y) = d(y,x)$
- $d(x,z) \leq d(x,y) + d(y,z)$
For $x \in X$ and $\varepsilon \in \mathbb{R}$ with $\varepsilon > 0$ , the open ball around $x$ of radius $\varepsilon$ is the set $B_\varepsilon(x) := \{y \in X \mid d(x,y) < \varepsilon\}$ . An open set in $X$ is a set which equals an arbitrary (possibly empty) union of open balls in $X$ , and $X$ together with these open sets forms a Hausdorff topological space. The topology on $X$ 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 $\bar{B}_\varepsilon(x) := \{y \in X \mid d(x,y) \leq \varepsilon\}$ is called a closed ball around $x$ of radius $\varepsilon$ . Every closed ball is a closed subset of $X$ in the metric topology.
The prototype example of a metric space is $\mathbb{R}$ itself, with the metric defined by $d(x,y) := |x-y|$ . 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 $d(x,y) = 0 \iff x=y$ without affecting the definition.
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"metric space" is owned by djao.
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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 393 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 92778 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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