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:
“A dense undergrowth of extension cords sustains my upper world of lights, music, and machines of comfort.”
—Mason Cooley (b. 1927)
“The monotonous dead clog me up and there is only
black done in black that oozes from the strongbox.
I must disembowel it and then set the heart, the legs,
of two who were one upon a large woodpile
and ignite, as I was once ignited, and let it whirl
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—Anne Sexton (1928–1974)