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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 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).