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,(X2i-X1i)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,|X2i-X1i|)

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,|X2i-X1i|)