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# Dense

A set in a first-countable space is dense in if , where is the set of limit points of . For example, the rational numbers are dense in the reals. In general, a subset of is dense if its set closure .

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

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

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## References

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.

Dense

## Cite this as:

Weisstein, Eric W. "Dense." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Dense.html