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)
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.