In topology and related areas of mathematics, a subset A of a topological space X is said to be dense in X if every point of X either belongs to A or else is arbitrarily "close" to a member of A — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine approximation). Formally, is dense in if the smallest closed subset of containing is itself.[1]

The density of a topological space is the least cardinality of a dense subset of

DefinitionEdit

A subset   of a topological space   is said to be a dense subset of   if any of the following equivalent conditions are satisfied:

  1. The smallest closed subset of   containing   is   itself.
  2. The closure of   in   is equal to   That is,  
  3. The interior of the complement of   is empty. That is,  
  4. Every point in   either belongs to   or is a limit point of  
  5. For every   every neighborhood   of   intersects   that is,  
  6.   intersects every non-empty open subset of  

and if   is a basis of open sets for the topology on   then this list can be extended to include:

  1. For every   every basic neighborhood   of   intersects  
  2.   intersects every non-empty  

Density in metric spacesEdit

An alternative definition of dense set in the case of metric spaces is the following. When the topology of   is given by a metric, the closure   of   in   is the union of   and the set of all limits of sequences of elements in   (its limit points),

 

Then   is dense in   if

 

If   is a sequence of dense open sets in a complete metric space,   then   is also dense in   This fact is one of the equivalent forms of the Baire category theorem.

ExamplesEdit

The real numbers with the usual topology have the rational numbers as a countable dense subset which shows that the cardinality of a dense subset of a topological space may be strictly smaller than the cardinality of the space itself. The irrational numbers are another dense subset which shows that a topological space may have several disjoint dense subsets (in particular, two dense subsets may be each other's complements), and they need not even be of the same cardinality. Perhaps even more surprisingly, both the rationals and the irrationals have empty interiors, showing that dense sets need not contain any non-empty open set. The intersection of two dense open subsets of a topological space is again dense and open.[proof 1] The empty set is a dense subset of itself. But every dense subset of a non-empty space must also be non-empty.

By the Weierstrass approximation theorem, any given complex-valued continuous function defined on a closed interval   can be uniformly approximated as closely as desired by a polynomial function. In other words, the polynomial functions are dense in the space   of continuous complex-valued functions on the interval   equipped with the supremum norm.

Every metric space is dense in its completion.

PropertiesEdit

Every topological space is a dense subset of itself. For a set   equipped with the discrete topology, the whole space is the only dense subset. Every non-empty subset of a set   equipped with the trivial topology is dense, and every topology for which every non-empty subset is dense must be trivial.

Denseness is transitive: Given three subsets   and   of a topological space   with   such that   is dense in   and   is dense in   (in the respective subspace topology) then   is also dense in  

The image of a dense subset under a surjective continuous function is again dense. The density of a topological space (the least of the cardinalities of its dense subsets) is a topological invariant.

A topological space with a connected dense subset is necessarily connected itself.

Continuous functions into Hausdorff spaces are determined by their values on dense subsets: if two continuous functions   into a Hausdorff space   agree on a dense subset of   then they agree on all of  

For metric spaces there are universal spaces, into which all spaces of given density can be embedded: a metric space of density   is isometric to a subspace of   the space of real continuous functions on the product of   copies of the unit interval. [2]

Related notionsEdit

A point   of a subset   of a topological space   is called a limit point of   (in  ) if every neighbourhood of   also contains a point of   other than   itself, and an isolated point of   otherwise. A subset without isolated points is said to be dense-in-itself.

A subset   of a topological space   is called nowhere dense (in  ) if there is no neighborhood in   on which   is dense. Equivalently, a subset of a topological space is nowhere dense if and only if the interior of its closure is empty. The interior of the complement of a nowhere dense set is always dense. The complement of a closed nowhere dense set is a dense open set. Given a topological space   a subset   of   that can be expressed as the union of countably many nowhere dense subsets of   is called meagre. The rational numbers, while dense in the real numbers, are meagre as a subset of the reals.

A topological space with a countable dense subset is called separable. A topological space is a Baire space if and only if the intersection of countably many dense open sets is always dense. A topological space is called resolvable if it is the union of two disjoint dense subsets. More generally, a topological space is called κ-resolvable for a cardinal κ if it contains κ pairwise disjoint dense sets.

An embedding of a topological space   as a dense subset of a compact space is called a compactification of  

A linear operator between topological vector spaces   and   is said to be densely defined if its domain is a dense subset of   and if its range is contained within   See also Continuous linear extension.

A topological space   is hyperconnected if and only if every nonempty open set is dense in   A topological space is submaximal if and only if every dense subset is open.

If   is a metric space, then a non-empty subset   is said to be  -dense if

 

One can then show that   is dense in   if and only if it is ε-dense for every  

See alsoEdit

ReferencesEdit

  1. ^ Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, ISBN 0-486-68735-X
  2. ^ Kleiber, Martin; Pervin, William J. (1969). "A generalized Banach-Mazur theorem". Bull. Austral. Math. Soc. 1 (2): 169–173. doi:10.1017/S0004972700041411.

proofs

  1. ^ Suppose that   and   are dense open subset of a topological space   If   then the conclusion that the open set   is dense in   is immediate, so assume otherwise. Let   is a non-empty open subset of   so it remains to show that   is also not empty. Because   is dense in   and   is a non-empty open subset of   their intersection   is not empty. Similarly, because   is a non-empty open subset of   and   is dense in   their intersection   is not empty.  

General referencesEdit