In mathematics, a subset of a topological space is called nowhere dense[1][2] or rare[3] if its closure has empty interior.
In a very loose sense, it is a set whose elements are not tightly clustered (as defined by the topology on the space) anywhere.
Meagre sets play an important role in the formulation of the Baire category theorem, which is used in the proof of several fundamental results of functional analysis.
Density nowhere can be characterized in different (but equivalent) ways.
The simplest definition is the one from density: A subset
is not dense in any nonempty open subset
Expanding out the negation of density, it is equivalent that each nonempty open set
contains a nonempty open subset disjoint from
[4] It suffices to check either condition on a base for the topology on
The second definition above is equivalent to requiring that the closure,
cannot contain any nonempty open set.
The notion of nowhere dense set is always relative to a given surrounding space.
Notably, a set is always dense in its own subspace topology.
However the following results hold:[10][11] A set is nowhere dense if and only if its closure is.
In general they do not form a 𝜎-ideal, as meager sets, which are the countable unions of nowhere dense sets, need not be nowhere dense.
is nowhere dense if and only if it is a subset of the boundary of some open set (for example the open set can be taken as the exterior of
A nowhere dense set is not necessarily negligible in every sense.
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 another example (a variant of the Cantor set), remove from
in lowest terms for positive integers
the nowhere dense set remaining after all such intervals have been removed has measure of at least
because of overlaps[17]) 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
Generalizing this method, one can construct in the unit interval nowhere dense sets of any measure less than
having infinite Lebesgue measure that is also nowhere dense in
of finite Lebesgue measure is commonly constructed when proving that the Lebesgue measure of the rational numbers
Taking the union of closed, rather than open, intervals produces the F𝜎-subset
Lebesgue measure that is also a nonmeager subset of
in this example can be replaced by any countable dense subset of