A set 
is discrete in a larger topological space 
if every point 
has a neighborhood 
such that 
. The points of 
are then said to be isolated
(Krantz 1999, p. 63). Typically, a discrete set is either finite or countably
infinite. For example, the set of integers is discrete on the real
line. Another example of an infinite discrete set is the set 
. On any reasonable space, a finite
set is discrete. A set is discrete if it has the discrete
topology, that is, if every subset is open.
In the case of a subset 
,
as in the examples above, one uses the relative
topology on 
.
Sometimes a discrete set is also closed. Then there cannot be any accumulation
points of a discrete set. On a compact set such
as the sphere, a closed discrete set must be finite because
of this.
