Dense Set

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. 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.

Read more about Dense Set:  Density in Metric Spaces, Examples, Properties, Related Notions

Famous quotes containing the words dense and/or set:

    While you’re playing cards with a regular guy or having a bite to eat with him, he seems a peaceable, good-humoured and not entirely dense person. But just begin a conversation with him about something inedible, politics or science, for instance, and he ends up in a deadend or starts in on such an obtuse and base philosophy that you can only wave your hand and leave.
    Anton Pavlovich Chekhov (1860–1904)

    A fool, A fool! I met a fool i’ the forest,
    A motley fool. A miserable world!
    As I do live by food, I met a fool,
    Who laid him down and basked him in the sun,
    And railed on Lady Fortune in good terms,
    In good set terms, and yet a motley fool.
    William Shakespeare (1564–1616)