In topology, 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 an 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 a limit point.

A set which is made up only of isolated points is called a discrete set, e.g., finite set. A discrete subset of Euclidean space is countable; however, a set can be countable but not discrete, e.g. the rational numbers.

A closed set with no isolated point is called a perfect set.

Examples

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

• For the set $S=\backslash \{0\backslash \}\backslash cup2$, the point 0 is an isolated point.

• For the set $S=\backslash \{0\backslash \}\backslash cup\; \backslash \{1,\; 1/2,\; 1/3,\; \backslash dots\; \backslash \}$, 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 N={0, 1, 2, ...} of natural numbers is a discrete set.

See also

• discrete space
• freely discontinuous
• free regular set

General topology

Isolierter Punkt | Izolita punkto | Point isolé | Punto isolato | 孤点