Density (disambiguation)

Density and dense usually refer to a measure of how much of some entity is within a fixed amount of space. Types of density include:

Physics

Density of mass

Densities of other entities

Dense set

In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if every point x in X either belongs to A or is a limit point of A. Informally, for every point in X, the point is either in A or arbitrarily "close" to a member of A — for instance, every real number is either a rational number or has one arbitrarily close to it (see Diophantine approximation).

Formally, a subset A of a topological space X is dense in X if for any point x in X, any neighborhood of x contains at least one point from A (i.e., A has non-empty intersection with every non-empty open subset of X). Equivalently, A is dense in X if and only 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.

The density of a topological space X is the least cardinality of a dense subset of X.

Density in metric spaces

An alternative definition of dense set in the case of metric spaces is the following. When the topology of X is given by a metric, the closure of A in X is the union of A and the set of all limits of sequences of elements in A (its limit points),

Pseudocomplement

In mathematics, particularly in order theory, a pseudocomplement is one generalization of the notion of complement. In a lattice L with bottom element 0, an element x ∈ L is said to have a pseudocomplement if there exists a greatest element x* ∈ L, disjoint from x, with the property that x ∧ x* = 0. More formally, x* = max{ y ∈ L | x ∧ y = 0 }. The lattice L itself is called a pseudocomplemented lattice if every element of L is pseudocomplemented. Every pseudocomplemented lattice is necessarily bounded, i.e. it has a 1 as well. Since the pseudocomplement is unique by definition (if it exists), a pseudocomplemented lattice can be endowed with a unary operation * mapping every element to its pseudocomplement; this structure is sometimes called a p-algebra. However this latter term may have other meanings in other areas of mathematics.

Properties

In a p-algebra L, for all x, y ∈ L:

Dense (film)

Dense is a 2004 American television short film directed and written by Vanessa Williams and Shari Poindexter, and aired on the TV channel Showtime.

Plot

A man gives his girlfriend an ultimatum: to drop her career to start a family or he will leave her. She begins to question the relationship realizing maybe she was dense.

Cast

Festival screenings

References

External links