A set S is discrete in a larger topological space X if every point x element S has a neighborhood U such that S intersection U = {x}. The points of S are then said to be isolated. Typically, a discrete set is either finite or countably infinite. For example, the set of integers is discrete on the real line. Another example of an infinite discrete set is the set {1/n for all integers n>1}. On any reasonable space, a finite set is discrete. A set is discrete if it has the discrete topology, that is, if every subset is open. In the case of a subset S, as in the examples above, one uses the relative topology on S. Sometimes a discrete set is also closed. Then there cannot be any accumulation points of a discrete set. On a compact set such as the sphere, a closed discrete set must be finite because of this.

accumulation point | compact space | discrete topology | isolated point | neighborhood | topological space