Difference between revisions of "Metric (analysis)"
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Together, the set <math>S</math> and the metric <math>d</math> form a [[metric space]]. | Together, the set <math>S</math> and the metric <math>d</math> form a [[metric space]]. | ||
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+ | 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 <math>B_{x,\epsilon}:=\{y\in S:d(x,y)<\epsilon\}</math>). | ||
==Common metrics== | ==Common metrics== |
Latest revision as of 20:15, 13 October 2019
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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