Open Balls in a Metric Space
The metric that makes a metric space is not a fixed object. A variety of metrics may be defined on a space, and produce topologically equivalent spaces – even if the distance between the same points is different for different metrics.
Four commonly used metrics in setr^2 are:
The open ball
is defined as the set of points satisfying
and the boundary of the ball is
The boundary of the open ball (not necessarily an actual ball)
of radius 1 defined by the metric
at the point
for each metric above is given below (except for
The interior of
is the single point
