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Nowhere Dense

A set is said to be nowhere dense if the interior of the set closure of is the empty set. For example, the Cantor set is nowhere dense.

There exist nowhere dense sets of positive measure. For example, enumerating the rationals in as ${q_n}$ and choosing an open interval of length containing for each , then the union of these intervals has measure at most 1/2. Hence, the set of points in but not in any of ${I_n}$ has measure at least 1/2, despite being nowhere dense.

Portions of this entry contributed by Dave Milovich

REFERENCES:

Ferreirós, J. "Lipschitz and Hankel on Nowhere Dense Sets and Integration." §5.2 in Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics. Basel, Switzerland: Birkhäuser, pp. 154-156, 1999.

Rudin, W. Functional Analysis, 2nd ed. New York: McGraw-Hill, p. 42, 1991.

CITE THIS AS:

Milovich, Dave and Weisstein, Eric W. "Nowhere Dense." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/NowhereDense.html