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In mathematics, a subset of a topological space is called nowhere dense or rare if its closure has empty interior.The order of operations is important. For example, the set of rational numbers, as a subset of the real numbers, , has the property that its ''interior'' has an empty , but it is not nowhere dense; in fact it is dense in . In a very loose sense, it is a set whose elements are not tightly clustered (as defined by the topology on the space) anywhere. For example, the integers are nowhere dense among the reals, whereas an open ball is not. The surrounding space matters: a set may be nowhere dense when considered as a subset of a topological space , but not when considered as a subset of another topological space . Notably, a set is always dense in its own subspace topology. A countable union of nowhere dense sets is called a meagre set. Meager sets play an important role in the formulation of the Baire category theorem.


Characterizations


Density nowhere can be characterized three different (but equivalent ways). The simplest definition is the one from density:
A subset S of a topological space X is said to be ''dense'' in another set U if the intersection S \cap U is a dense subset of U. S is or in X if S is not dense in any open subset U of X.
Expanding out the negation of density, it is equivalent to require that each open set U contains an open subset disjoint from S. It suffices to check either condition on a base for the topology on X, and density nowhere in \mathbb is often described as being dense in no open interval.


Definition by closure


The second definition above is equivalent to requiring that the closure, \operatorname_X S, cannot contain any open set. This is the same as saying that the interior of the closure of S (both taken in X) is empty; that is,
\operatorname_X \left(\operatorname_X S\right) = \varnothing.
Alternatively, the complement of the closure X \setminus \left(\operatorname_X S\right) must be a dense subset of X.


Definition by boundaries


From the previous remark, S is nowhere dense in X if and only if S is a subset of the boundary of a dense open subset: namely, X \setminus \left(\operatorname_X S\right). In fact, one can remove the denseness condition:
S is nowhere dense iff there exists some open subset U of X such that S \subseteq \operatorname_X U,
Alternatively, one can strengthen the containment to equality by taking the closure:
S is nowhere dense iff there exists some open subset U of X such that \operatorname_X S = \operatorname_X U.
If S is closed, this implies by trichotomy that S is nowhere dense if and only if S is equal to its topological boundary.


Properties and sufficient conditions


  • A set is nowhere dense iff its closure is. Thus a nowhere dense set need not be closed (for instance, the set is nowhere dense in the reals), but is properly contained in a nowhere dense closed set.
  • Suppose A \subseteq B \subseteq X. * If A is nowhere dense in B then A is nowhere dense in X. * If A is nowhere dense in X and B is an open subset of X then A is nowhere dense in B.
  • Every subset of a nowhere dense set is nowhere dense.
  • The union of finitely many nowhere dense sets is nowhere dense.
Thus the nowhere dense sets form an ideal of sets, a suitable notion of negligible set. The union of countably many nowhere dense sets, however, need not be nowhere dense. (Thus, the nowhere dense sets do not, in general, form a sigma-ideal.) Instead, such a union is called a meagre set or a set of first category.


Examples


  • is nowhere dense in : although the points get arbitrarily close to 0, the closure of the set is , which has empty interior (and is thus also nowhere dense in ).
  • is nowhere dense in .
  • is nowhere dense in but the rationals are not (they are dense everywhere).
  • is nowhere dense in : it is dense in the interval , and in particular the interior of its closure is .
  • The empty set is nowhere dense. In a discrete space, the empty set is the ''only'' such subset.
  • In a T1 space, any singleton set that is not an isolated point is nowhere dense.
  • The boundary of every open set and of every closed set is nowhere dense.
  • A vector subspace of a topological vector space is either dense or nowhere dense.



Nowhere dense sets with positive measure


A nowhere dense set is not necessarily negligible in every sense. For example, if X is the unit interval , not only is it possible to have a dense set of Lebesgue measure zero (such as the set of rationals), but it is also possible to have a nowhere dense set with positive measure. For one example (a variant of the Cantor set), remove from ,1all dyadic fractions, i.e. fractions of the form in lowest terms for positive integers and , and the intervals around them: , . Since for each this removes intervals adding up to at most , the nowhere dense set remaining after all such intervals have been removed has measure of at least (in fact just over 0.535... because of overlaps) and so in a sense represents the majority of the ambient space . This set is nowhere dense, as it is closed and has an empty interior: any interval is not contained in the set since the dyadic fractions in have been removed. Generalizing this method, one can construct in the unit interval nowhere dense sets of any measure less than , although the measure cannot be exactly 1 (else the complement of its closure would be a nonempty open set with measure zero, which is impossible).


See also


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Notes





References





Bibliography


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External links



Some nowhere dense sets with positive measure
Category:General topology de:Dichte Teilmenge#Nirgends dichte Teilmenge