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

**R**^{n}, 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 (

E^{n}):

**Pythagorean Metric Space**
Results from using the distance rule

d = sqrt (sum (i=1..

*n*,(X

_{2}^{i}-X

_{1}^{i})

^{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

_{2}^{i}-X

_{1}^{i}|)

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

_{2}^{i}-X

_{1}^{i}|)