Nowhere Dense
المؤلف:
Ferreirós, J.
المصدر:
"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.
الجزء والصفحة:
...
12-1-2022
1695
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 ![[0,1]](https://mathworld.wolfram.com/images/equations/NowhereDense/Inline3.svg)
as ![<span style=]()
{q_n}" src="https://mathworld.wolfram.com/images/equations/NowhereDense/Inline4.svg" style="height:22px; width:30px" /> 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 ![[0,1]](https://mathworld.wolfram.com/images/equations/NowhereDense/Inline9.svg)
but not in any of ![<span style=]()
{I_n}" src="https://mathworld.wolfram.com/images/equations/NowhereDense/Inline10.svg" style="height:22px; width:27px" /> 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.
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