metric space
A metric space is a set $X$ together with a real valued function $d:X\times X\u27f6\mathbb{R}$ (called a metric, or sometimes a distance function) such that, for every $x,y,z\in X$,

•
$d(x,y)\ge 0$, with equality^{1}^{1}This condition can be replaced with the weaker statement $d(x,y)=0\iff x=y$ without affecting the definition. if and only if $x=y$

•
$d(x,y)=d(y,x)$

•
$d(x,z)\le d(x,y)+d(y,z)$
For $x\in X$ and $\epsilon \in \mathbb{R}$ with $\epsilon >0$, the open ball around $x$ of radius $\epsilon $ is the set $$. 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 ${\overline{B}}_{\epsilon}(x):=\{y\in X\mid d(x,y)\le \epsilon \}$ is called a closed ball around $x$ of radius $\epsilon $. 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 $d(x,y):=xy$. 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  20130322 11:53:19 
Last modified on  20130322 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 8200 
Classification  msc 8300 
Classification  msc 8100 
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 