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In mathematics, a point x is called an isolated point of a subset S (in a topological space X) if x is an element of S but there exists a neighborhood of x which does not contain any other points of S. This is equivalent to saying that the singleton {x} is an open set in the topological space S (considered as a subspace of X). If the space X is a Euclidean space (or any other metric space), then x is an isolated point of S if there exists an open ball around x which contains no other points of S. (Introducing the notion of sequences and limits, one can say equivalently that an element x of S is an isolated point of S if and only if it is not a limit point of S.)

Discrete set

A set that is made up only of isolated points is called a discrete set (see also discrete space). Any discrete subset S of Euclidean space must be countable, since the isolation of each of its points together with the fact that rationals are dense in the reals means that the points of S may be mapped into a set of points with rational coordinates, of which there are only countably many. However, not every countable set is discrete, of which the rational numbers under the usual Euclidean metric are the canonical example.

A set with no isolated point is said to be dense-in-itself (every neighbourhood of a point contains other points of the set). A closed set with no isolated point is called a perfect set (it has all its limit points and none of them are isolated from it).

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.

Standard examples

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