Metric space
A metric space
- Symmetry:
for all - Triangle inequality:
for all - Positive definite:
iff.
It immediately follows that
Examples
The quintessential example is the pythagorean distance function on euclidean space, which in one dimension is simply the difference
A trivial example is the discrete metric, which yields the Discrete topology.
Properties
- Reverse triangle inequality
- Metric spaces induce a Metric topology with the open balls as its Topological basis. Thus a metric space gives the most intuitive definition of open and closed sets, which is generalised by a topological space.