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**