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Discrete Metric

The metric g defined on a nonempty set X by

g(x,x)=0
(1)
g(x,y)=1
(2)

if x!=y for all x,y in X.

It follows that the open ball of radius r>0 and center at x_0

B(x_0,r)={x in X|g(x_0,x)<r}
(3)

is

{{x_0} if 0<r<=1; X if r>1.
(4)

The metric topology induced is the discrete topology.

This entry contributed by Margherita Barile

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Cite this as:

Barile, Margherita. "Discrete Metric." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/DiscreteMetric.html

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