Isolated point - Definition and Overview

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.

A set which is made up only of isolated points is called a discrete set.

Examples

• For the set <math>S=\{0\}\cup [1, 2]<math>, the point 0 is an isolated point.

• For the set <math>S=\{0\}\cup \{1, 1/2, 1/3, \dots \}<math>, 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