A set A in a first-countable space is dense in B if B=A union L, where L is the set of limit points of A. For example, the rational numbers are dense in the reals. In general, a subset A of X is dense if its set closure cl(A)=X.

A real number alpha is said to be b-dense iff, in the base-b expansion of alpha, every possible finite string of consecutive digits appears. If alpha is b-normal, then alpha is also b-dense. If, for some b, alpha is b-dense, then alpha is irrational. Finally, alpha is b-dense iff the sequence {b^nalpha} is dense (Bailey and Crandall 2001, 2003).

See also

Derived Set, Natural Invariant, Normal Number, Nowhere Dense, Perfect Set, Set Closure

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Bailey, D. H. and Crandall, R. E. "On the Random Character of Fundamental Constant Expansions." Exper. Math. 10, 175-190, 2001.Bailey, D. H. and Crandall, R. E. "Random Generators and Normal Numbers." Exper. Math. 11, 527-546, 2002.

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Cite this as:

Weisstein, Eric W. "Dense." From MathWorld--A Wolfram Web Resource.

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