Take any set of points X. We can measure distance between two points x and y in X by a distance function d : X → R where d satisfies the following properties:

M1 0 ≤ d(x,y) < ∞

M2 d(x,y) = 0 iff x = y non-negativity property

M3 d(x,y) = d(y,x) symmetric property

M4 d(x,y) ≤ d(x,z) + d(z,y) sub-additive or triangle inequality

If d satisfies M1 - M4 d is called a Metric and (x,d) is called a Metric Space

If d satisfies M1, M3 and M4 (X,d) is called a Psuedo or Semi Metric Space

If d satisfies M1, M2 and M4 (X,d) is called a Quasi Metric Space
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|)

Slightly more obscure (although in some ways, more important) examples include:

Discrete Metric Space
formed by taking any set and the distance function:
d(x,y) = 0 if x = y
d(x,y) = 1 if x != y
often used in pathological examples to prove various theorems and statements wrong.

Induced Metric
If A is a subset of X and (X,d) is a metric space, then we call (A,d) the induced metric on A. Whilst this seems obvious (if you do Math, anyway) it's very useful in proving theorems in other areas, such as Topology.

There is a tiny error in the writeup above by r2001, and it presents the occasion for a cute little proof. M1 follows from M2, M3 and M4 as follows:

0 = d(x,x) from M2
d(x,x) < = d(x,y) + d(y,x) from M4
d(x,y) + d(y,x) = d(x,y) + d(x,y) = 2d(x,y) from M3
so...

0 = d(x,x) < = 2d(x,y)
and thus 0 < = d(x,y).

So 'd' need only satisfy M2, M3, and M4 if it is to be a metric.

Of course, only a mathematician would think to take z = y in M4....

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