, a branch of mathematics
, a point x
of a set S
is called an isolated point
if there exists a neighborhood
not containing other points of S
In particular, in a Euclidean space
(or in a metric space
is an isolated point of S
, if one can find an open ball
which contains no other points of S
Equivalently, a point x
is not isolated if and only if x
is an accumulation point
A set which is made up only of isolated points is called a discrete set. A discrete subset of Euclidean space is countable; however, a set can be countable but not discrete, e.g. the rational numbers. See also discrete space.
A closed set with no isolated point is called a perfect set.
The number of isolated points is a topological invariant, i.e. if two topological spaces and are homeomorphic, the number of isolated points in each is equal.
Topological spaces in the following examples are considered as subspaces of the real line.
- For the set , the point 0 is an isolated point.
- For the set , each of the points 1/k is an isolated point, but 0 is not an isolated point because there are other points in S as close to 0 as desired.
- The set of natural numbers is a discrete set.