Isolated point

Isolated point

In topology, a branch of mathematics, a point x of a set S is called an isolated point, if there exists a neighborhood of x not containing other points of S. In particular, in a Euclidean space (or in a metric space), x is an isolated point of S, if one can find an open ball around x 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 X and Y 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 S={0}cup [1, 2], the point 0 is an isolated point.
  • For the set S={0}cup {1, 1/2, 1/3, dots }, 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 {mathbb N} = {0, 1, 2, ldots } of natural numbers is a discrete set.

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