# metric space

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 equality11This condition can be replaced with the weaker statement $d(x,y)=0\iff x=y$ without affecting the definition. 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 (http://planetmath.org/PseudometricTopology)).

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.

## References

• 1 J.L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955.
 Title metric space Canonical name MetricSpace Date of creation 2013-03-22 11:53:19 Last modified on 2013-03-22 11:53:19 Owner djao (24) Last modified by djao (24) Numerical id 15 Author djao (24) Entry type Definition Classification msc 54E35 Classification msc 82-00 Classification msc 83-00 Classification msc 81-00 Related topic Neighborhood Related topic VectorNorm Related topic T2Space Related topic Ultrametric Related topic QuasimetricSpace Related topic NormedVectorSpace Related topic PseudometricSpace Defines distance metric Defines metric Defines distance Defines metric topology Defines open ball Defines closed ball