Discrete Set

A set S is discrete in a larger topological space X if every point x in S has a neighborhood U such that S intersection U={x}. The points of S are then said to be isolated (Krantz 1999, p. 63). 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.

See also

Accumulation Point, Compact Space, Discrete Topology, Isolated Point, Neighborhood, Topological Space

Explore with Wolfram|Alpha


Krantz, S. G. "Discrete Sets and Isolated Points." §4.6.2 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 63-64, 1999.

Referenced on Wolfram|Alpha

Discrete Set

Cite this as:

Weisstein, Eric W. "Discrete Set." From MathWorld--A Wolfram Web Resource.

Subject classifications