Metric (analysis)
(Redirected from Metric (set theory))
A metric on a set
is a function
which obeys the following three properties:
- Symmetry:
for all points
.
- Positivity:
for all
and
if and only if
.
- The triangle inequality:
for all
.
Together, the set and the metric
form a metric space.
Every metric space can be used to form a topology by considering taking the set of open balls as a topological basis (i.e. the sets ).
Common metrics
- For
, the Euclidean metric
is the conventional distance function.
- For any set
, the discrete metric
and
otherwise.
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