metric space

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

• $d(x,y)\ge 0$, with equality 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 $\epsilon \in ℝ$ 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 $ℝ$ 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.