Nowhere dense set

In mathematics, a subset of a topological space is called nowhere dense or rare^[1] if its closure has empty interior.^{[note 1]} 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 ${\displaystyle A}$ may be nowhere dense when considered as a subset of a topological space ${\displaystyle X,}$ but not when considered as a subset of another topological space ${\displaystyle Y.}$ Notably, a set is always dense in its own subspace topology.

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.

Characterizations[]

Density nowhere can be characterized in three different (but equivalent) ways. The simplest definition is the one from density:

A subset ${\displaystyle S}$ of a topological space ${\displaystyle X}$ is said to be dense in another set ${\displaystyle U}$ if the intersection ${\displaystyle S\cap U}$ is a dense subset of ${\displaystyle U.}$ ${\displaystyle S}$ is nowhere dense or rare in ${\displaystyle X}$ if ${\displaystyle S}$ is not dense in any nonempty open subset ${\displaystyle U}$ of ${\displaystyle X.}$

Expanding out the negation of density, it is equivalent to require that each nonempty open set ${\displaystyle U}$ contains a nonempty open subset disjoint from ${\displaystyle S.}$^[2] It suffices to check either condition on a base for the topology on ${\displaystyle X,}$ and density nowhere in ${\displaystyle \mathbb {R} }$ is often described as being dense in no open interval.^[3][4]

Definition by closure[]

The second definition above is equivalent to requiring that the closure, ${\displaystyle \operatorname {cl} _{X}S,}$ cannot contain any nonempty open set.^[5] This is the same as saying that the interior of the closure of ${\displaystyle S}$ (both taken in ${\displaystyle X}$) is empty; that is,

${\displaystyle \operatorname {int} _{X}\left(\operatorname {cl} _{X}S\right)=\varnothing .}$^[6][7]

Alternatively, the complement of the closure ${\displaystyle X\setminus \left(\operatorname {cl} _{X}S\right)}$ must be a dense subset of ${\displaystyle X.}$^[2][6]

Definition by boundaries[]

From the previous remark, ${\displaystyle S}$ is nowhere dense in ${\displaystyle X}$ if and only if ${\displaystyle S}$ is a subset of the boundary of a dense open subset: namely, ${\displaystyle X\setminus \left(\operatorname {cl} _{X}S\right).}$ In fact, one can remove the denseness condition:

${\displaystyle S}$ is nowhere dense if and only if there exists some open subset ${\displaystyle U}$ of ${\displaystyle X}$ such that ${\displaystyle S\subseteq \operatorname {bd} _{X}U,}$

Alternatively, one can strengthen the containment to equality by taking the closure:

${\displaystyle S}$ is nowhere dense if and only if there exists some open subset ${\displaystyle U}$ of ${\displaystyle X}$ such that ${\displaystyle \operatorname {cl} _{X}S=\operatorname {bd} _{X}U.}$^[8]

If ${\displaystyle S}$ is closed, this implies by trichotomy that ${\displaystyle S}$ is nowhere dense if and only if ${\displaystyle S}$ is equal to its topological boundary.^[1]

Properties and sufficient conditions[]

• A set is nowhere dense if and only if its closure is.^[1] Thus a nowhere dense set need not be closed (for instance, the set ${\displaystyle \left\{{\frac {1}{n}}:n\in \mathbb {N} \right\}}$ is nowhere dense in the reals), but is then properly contained in a nowhere dense closed set.
• Suppose ${\displaystyle A\subseteq B\subseteq X.}$
  • If ${\displaystyle A}$ is nowhere dense in ${\displaystyle B}$ then ${\displaystyle A}$ is nowhere dense in ${\displaystyle X.}$
  • If ${\displaystyle A}$ is nowhere dense in ${\displaystyle X}$ and ${\displaystyle B}$ is an open subset of ${\displaystyle X}$ then ${\displaystyle A}$ is nowhere dense in ${\displaystyle B.}$^[1]
• Every subset of a nowhere dense set is nowhere dense.^[9]
• The union of finitely many nowhere dense sets is nowhere dense.^[9]

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