Isolated point

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 and there exists a neighborhood of x that 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).

Another equivalent formulation is: an element x of S is an isolated point of S if and only if it is not a limit point of S. If the space X is a metric space, for example a Euclidean space, then an element x of S is an isolated point of S if there exists an open ball around x that contains only finitely many elements of S. A point set that is made up only of isolated points is called a discrete set or discrete point 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 injectively onto 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.

The number of isolated points is a topological invariant, i.e. if two topological spaces X, Y are homeomorphic, the number of isolated points in each is equal.

The Morse lemma states that non-degenerate critical points of certain functions are isolated.

Consider the set F of points x in the real interval (0,1) such that every digit xi of their binary representation fulfills the following conditions: Informally, these conditions means that every digit of the binary representation of

Now, F is an explicit set consisting entirely of isolated points but has the counter-intuitive property that its closure is an uncountable set.

, and let F be a set consisting of one point from each Ik.