Isolated point


In mathematics, a point x is called an isolated point of a subset S 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 is an open set in the topological space S. If the space X is a Euclidean space, then x is an isolated point of S if there exists an open ball around x which contains no other points of S.

Discrete set

A set that is made up only of isolated points is called a discrete set. Any discrete subset S of Euclidean space must be countable, since the isolation of each of its points together with the fact the 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. 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 and are homeomorphic, the number of isolated points in each is equal.

Standard examples

s in the following examples are considered as subspaces of the real line with the standard topology.
Consider the set of points in the real interval such that every digit of their binary representation fulfills the following conditions:
Now, is an explicit set consisting entirely of isolated points which has the counter-intuitive property that its closure is an uncountable set.
Another set with the same properties can be obtained as follows. Let be the middle-thirds Cantor set, let be the component intervals of, and let be a set consisting of one point from each. Since each contains only one point from, every point of is an isolated point. However, if is any point in the Cantor set, then every neighborhood of contains at least one, and hence at least one point of. It follows that each point of the Cantor set lies in the closure of, and therefore has uncountable closure.