, 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
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
Density nowhere can be characterized three different (but equivalent ways). The simplest definition is the one from density:
A subset of a topological space is said to be ''dense'' in another set if the intersection is a dense subset of . is or in if is not dense in any open subset of .
Expanding out the negation of density, it is equivalent to require that each open set
contains an open subset disjoint from
. It suffices to check either condition on a base
for the topology on
, and density nowhere in
is often described as being dense in no open interval
Definition by closure
The second definition above is equivalent to requiring that the closure,
, cannot contain any open set. This is the same as saying that the interior
of the closure
(both taken in
is empty; that is,
Alternatively, the complement of the closure
must be a dense subset of
Definition by boundaries
From the previous remark,
is nowhere dense in
if and only if
is a subset of the boundary of a dense open subset: namely,
. In fact, one can remove the denseness condition:
is nowhere dense iff there exists some open subset of such that ,
Alternatively, one can strengthen the containment to equality by taking the closure:
is nowhere dense iff there exists some open subset of such that .
is closed, this implies by trichotomy
is nowhere dense if and only if
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.
* If is nowhere dense in then is nowhere dense in
* If is nowhere dense in and is an open subset of then is nowhere dense in
- 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.
- 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
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 ,1
all dyadic fraction
s, 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).
Some nowhere dense sets with positive measure
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