Wikipedia / Nowhere dense set
Wikipedia / Nowhere dense set
In mathematics, a subset of a topological space is called nowhere dense or rare 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. For example, the integers are nowhere dense among the realss, whereas the interval (0, 1) is not nowhere dense.
A countable union of nowhere dense sets is called a meagre set. 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.
Definition
Density nowhere can be characterized in different (but equivalent) ways. The simplest definition is the one from density: of a topological space is said to be dense is a dense subset is or Expanding out the negation of density, it is equivalent that each nonempty open set It suffices to check either condition on a base is often described as being dense in no open interval.
Definition by closure
cannot contain any nonempty open set. This is the same as saying that the interior of the closure is empty; that is, in other words, the exterior
Properties
The notion of nowhere dense set has the subspace topology is nonempty, it will not be nowhere dense as a subset of itself. However the following results hold:
A set is nowhere dense if and only if its closure is.
Every subset of a nowhere dense set is nowhere dense, and a finite union of nowhere dense sets is nowhere dense. Thus the nowhere dense sets form an ideal of sets, a suitable notion of negligible set. In general they do not form a 𝜎-ideal, as meager sets
The boundary of every open set and of every closed set is closed and nowhere dense. A closed set is nowhere dense if and only if it is equal to its boundary, if and only if it is equal to the boundary of some open set 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).
Examples
since the closure has empty interior.
The Cantor set
are not (they are dense everywhere).
is
The empty set is nowhere dense. In a discrete space, the empty set is the nowhere dense set.
In a T1 space, any singleton set that is not an isolated point is nowhere dense.
A vector subspace of a topological vector space is either dense or nowhere dense.
Nowhere dense sets with positive measure
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. One such example is the Smith–Volterra–Cantor set.
For another example (a variant of the Cantor set all dyadic fractions because of overlaps have been removed.
although the measure cannot be exactly 1 (because otherwise the complement of its closure would be a nonempty open set with measure zero, which is impossible).
having finite Lebesgue measure This may be done by choosing any bijection to merely be a surjection (here, the Minkowski sum Taking the union of closed, rather than open, intervals produces the F-subset is a Baire space Lebesgue measure that is also a nonmeager subset is of the second category a comeager subset if and only if its