Each metric space has a
topological space closely associated with it, called the space's "
metric topology" (follow the link).
The most recognizable metric spaces are all based on
Rn, that is, the
Cartesian product of
n copies of the set of
real numbers. Not only that, all three of these metric spaces induce the same
topological space (
En):
Pythagorean Metric Space
Results from using the distance rule
d = sqrt (sum (i=1..
n,(X
2i-X
1i)
2)
that is, the familiar
Pythagorean Theorem generalized to
n dimensions.
Manhattan or Taxicab Space
results from using the distance rule
d = sum (i=1..
n,|X
2i-X
1i|)
This is the distance you would travel (perhaps in a
taxicab) between any two points in a rectangular street grid (such as
Manhattan nearly has).
Box Space somebody suggest a better name
results from using the distance rule
d = max (i=1..
n,|X
2i-X
1i|)