Mathematicians have a fascinating way with words --- or maybe vice versa. Who could imagine something which is both "dense" and "nowhere dense"? But that's exactly what the rational numbers (the set of all possible fractions, i.e. ratios of integers) are.

Rational numbers are "dense" because no matter how tiny a neighborhood you pick around any number, there are still an infinite number of fractions that lie in that neighborhood. Simultaneously, the rationals are "nowhere dense" because however tiny a zone you pick around a rational number, there are infinitely many non-rational numbers (irrationals, numbers not expressible as fractions) in that zone too.

An important part of the game of mathematics involves thinking up pathological or perverse examples and seeing where they lead. For instance, take the decimal numbers that don't have any odd digits in their representation (please!). Or consider the set of numbers left when you start with the interval between 0 and 1 and cut out the middle third --- and then cut the middle third out from each of the pieces that remain, and on, and on ad infinitum. Or how about the function sin(1/x) and how it behaves near x=0, where it wiggles faster and faster? Or ponder x*sin(1/x) near x=0, where the wiggles are getting squeezed down by the overall multiplier "x"?

The magic of these weird and wacky cases is that they stress our everyday understanding, reveal gaps in our knowledge, and break down our prejudices. That's not a bad goal for the nonmathematical parts of life either, eh?

Wednesday, May 12, 1999 at 21:10:14 (EDT) = 1999-05-12


No: a set cannot be both dense _and_ nowhere dense as these are fairly opposite in meaning:

A set E is nowhere dense in a set X iff the interior of its closure is empty (i.e its adherence contains no non-empty open sets).
The _order_ of these operations is important : for instance, the set of rational numbers' closure is the set of real numbers and the interior of that set is that set itself (therefore its not empty). On the other hand, the closure of the interior of the set of rational numbers is empty... Any finite set is nowhere dense in the set of real numbers. Actually, the set of all integers is also nowhere dense in IR.

A set E is said to be dense in a set X if the closure of E is X.
For example, the set of rational numbers is dense in the set of real numbers.

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