In mathematics, the term dense has at least two different meanings.
- A subset A of a topological space X is said to be dense 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. Equivalently, every nonempty open subset of X intersects A, or in other words: the interior of the complement of A is empty. As an example, the set of rational numbers is a dense subset of the real numbers.
- A partial order on a set S is said to be dense if, for all x and y in S for which x < y, there is a z in S such that x < z < y. The rational numbers with the ordinary ordering are a densely ordered set in this sense, as are the real numbers.
- A subset B of a partially ordered set A is dense in A if for any x < y in A, there is some z in B such that x < z < y. In case the order is a linear order, then B is dense in A in the present sense if and only if B is dense in the order topology on A. Hence the first two meanings above are related.
See also density in physics.