In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if, intuitively, any point in X can be "well-approximated" by points in A. Formally, A is dense in X if for any point x in X, any neighborhood of x contains at least a point from A. A Möbius strip, a surface with only one side and one edge; such shapes are an object of study in topology. ... Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. The relationship of one set being a subset of another is called inclusion. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. ...

Equivalently, A is dense in X if the only closed subset of X containing A is X itself. This can also be expressed by saying that the closure of A is X, or that the interior of the complement of A is empty. In topology and related branches of mathematics, a closed set is a set whose complement is open. ... In mathematics, the closure of a set S consists of all points which are intuitively close to S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior. ... In mathematics, the interior of a set S consists of all points which are intuitively not on the edge of S. A point which is in the interior of S is an interior point of S. The notion of interior is in many ways dual to the notion of closure. ...

An alternative definition in the case of metric spaces is the following: The set A in a metric space X is dense if every x in X is a limit of a sequence of elements in A. In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... Limit of a sequence is one of the oldest concepts in mathematical analysis. ...

Examples

• Every topological space is dense in itself.
• The real numbers with the usual topology have the rational numbers and the irrational numbers as dense subsets.
• A metric space M is dense in its completion γM.

In mathematics, the set of real numbers, denoted R, or in blackboard bold , is the set of all rational and irrational numbers. ... In mathematics, a rational number (commonly called a fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In mathematics, an irrational number is any real number that is not a rational number, i. ... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For...

See also

• density (disambiguation)
• separable space, a space with a countable dense subset
• nowhere dense set, the opposite notion