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Metric Space
A metric space is a set 
together with a real
valued function
(called a metric, or sometimes a distance function) such that, for every
,
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).
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.
Bibliography
- 1
- J.L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955.
