Dense set Information
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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
Definition
A subset of a topological space is said to be a dense subset of if any of the following equivalent conditions are satisfied:
- The smallest closed subset of containing is itself.
- The closure of in is equal to That is,
- The interior of the complement of is empty. That is,
- Every point in either belongs to or is a limit point of
- For every every neighborhood of intersects that is,
- 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:
- For every every basic neighborhood of intersects
- intersects every non-empty
Density in metric spaces
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.
Examples
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.
Properties
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 notions
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 also
- Blumberg theorem – Any real function on R admits a continuous restriction on a dense subset of R
- Dense order
- Dense (lattice theory)
References
- ^ Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, ISBN 0-486-68735-X
- ^ Kleiber, Martin; Pervin, William J. (1969). "A generalized Banach-Mazur theorem". Bull. Austral. Math. Soc. 1 (2): 169–173. doi: 10.1017/S0004972700041411.
proofs
- ^ 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 references
- Nicolas Bourbaki (1989) [1971]. General Topology, Chapters 1–4. Elements of Mathematics. Springer-Verlag. ISBN 3-540-64241-2.
- Bourbaki, Nicolas (1989) [1966]. General Topology: Chapters 1–4 [ Topologie Générale]. Éléments de mathématique. Berlin New York: Springer Science & Business Media. ISBN 978-3-540-64241-1. OCLC 18588129.
- Dixmier, Jacques (1984). General Topology. Undergraduate Texts in Mathematics. Translated by Berberian, S. K. New York: Springer-Verlag. ISBN 978-0-387-90972-1. OCLC 10277303.
- Munkres, James R. (2000). Topology (Second ed.). Upper Saddle River, NJ: Prentice Hall, Inc. ISBN 978-0-13-181629-9. OCLC 42683260.
- Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology ( Dover reprint of 1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 0507446
- Willard, Stephen (2004) [1970]. General Topology (First ed.). Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240.