Definitions

# 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.

## Examples

Topological spaces in the following examples are considered as subspaces of the real line.

• For the set $S=\left\{0\right\}cup \left[1, 2\right]$, the point 0 is an isolated point.
• For the set $S=\left\{0\right\}cup \left\{1, 1/2, 1/3, dots \right\}$, 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 $\left\{mathbb N\right\} = \left\{0, 1, 2, ldots \right\}$ of natural numbers is a discrete set.