, with equality11This condition can be replaced with the weaker statement without affecting the definition. if and only if
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 (http://planetmath.org/PseudometricTopology)).
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.
- 1 J.L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955.
|Date of creation||2013-03-22 11:53:19|
|Last modified on||2013-03-22 11:53:19|
|Last modified by||djao (24)|