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 ballis defined as the set of points satisfyingand the boundary of the ball is
The boundary of the open ball (not necessarily an actual ball)of radius 1 defined by the metricat the pointfor each metric above is given below (except forThe interior ofis the single point