Metric Spaces, The Distance Metric

The Distance Metric

A metric is a distance function on a set of points, mapping pairs of points into the nonnegative reals. We write |x,y| for the distance between x and y, and |x| for the distance between x and some fixed point, usually called the origin. Furthermore, the metric obeys three properties:

|x,y| = 0 iff x = y (isolation)

|x,y| = |y,x| (symmetry)

|x,y| + |y,z| ≥ |x,z| (triangular inequality)

Pseudo Metric

A pseudo metric may have |x,y| = 0, even when x ≠ y. For a given point x, all the points that are 0 distance from x are 0 distance from each other by the triangular inequality. Thus each point is a representative of a set of points, all zero distance apart, all occupying the same location in our topological space. Clump all these points together into a single point. The distance between two separate clumps is the distance between a point in the first and a point in the second. Use the triangular inequality to show this is well defined; it doesn't matter which point is selected on either side. The quotient set, the set of zero-distance aggregates, now has a valid metric.

Map the first space onto the second by carrying each point to its zero-distance cluster. Note that open balls in both worlds correspond, hence the function is bicontinuous and onto, giving a quotient space.

We usually assume this "clumping" has already been accomplished, whence the points of a metric space are topologically distinct.