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  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  has measure at least 1/2, despite being nowhere dense.

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.