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)
“Thus all probable reasoning is nothing but a species of sensation. ‘Tis not solely in poetry and music, we must follow our taste and sentiment, but likewise in philosophy, When I am convinc’d of any principle, ‘tis only an idea which strikes more strongly upon me. When I give the preference to one set of arguments above another, I do nothing but decide from my feeling concerning the superiority of their influence.”
—David Hume (1711–1776)