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In
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...
and related areas of mathematics, a
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
''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 number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s because every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine approximation). Formally, A is dense in X if the smallest closed subset of X containing A is X itself. The of a topological space X is the least
cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
of a dense subset of X.


Definition

A subset A of a topological space X is said to be a of X if any of the following equivalent conditions are satisfied:
  1. The smallest closed subset of X containing A is X itself.
  2. The closure of A in X is equal to X. That is, \operatorname_X A = X.
  3. The
    interior Interior may refer to: Arts and media * ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas * ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck * ''The Interior'' (novel), by Lisa See * Interior de ...
    of the complement of A is empty. That is, \operatorname_X (X \setminus A) = \varnothing.
  4. Every point in X either belongs to A or is a limit point of A.
  5. For every x \in X, every neighborhood U of x intersects A; that is, U \cap A \neq \varnothing.
  6. A intersects every non-empty open subset of X.
and if \mathcal is a basis of open sets for the topology on X then this list can be extended to include:
  1. For every x \in X, every neighborhood B \in \mathcal of x intersects A.
  2. A intersects every non-empty B \in \mathcal.


Density in metric spaces

An alternative definition of dense set in the case of metric spaces is the following. When the
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...
of X is given by a metric, the closure \overline of A in X is the union of A and the set of all limits of sequences of elements in A (its ''limit points''), \overline = A \cup \left\ Then A is dense in X if \overline = X. If \left\ is a sequence of dense open sets in a complete metric space, X, then \bigcap^_ U_n is also dense in X. This fact is one of the equivalent forms of the Baire category theorem.


Examples

The
real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s with the usual topology have the rational numbers as a countable dense subset which shows that the
cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
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.Suppose that A and B are dense open subset of a topological space X. If X = \varnothing then the conclusion that the open set A \cap B is dense in X is immediate, so assume otherwise. Let U is a non-empty open subset of X, so it remains to show that U \cap (A \cap B) is also not empty. Because A is dense in X and U is a non-empty open subset of X, their intersection U \cap A is not empty. Similarly, because U \cap A is a non-empty open subset of X and B is dense in X, their intersection U \cap A \cap B is not empty. \blacksquare 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 In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
continuous function defined on a closed interval
, b The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
/math> can be uniformly approximated as closely as desired by a polynomial function. In other words, the polynomial functions are dense in the space C
, b The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
/math> of continuous complex-valued functions on the interval
, b The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
equipped with the
supremum norm In mathematical analysis, the uniform norm (or ) assigns to real- or complex-valued bounded functions defined on a set the non-negative number :\, f\, _\infty = \, f\, _ = \sup\left\. This norm is also called the , the , the , or, when ...
. Every metric space is dense in its completion.


Properties

Every topological space is a dense subset of itself. For a set X equipped with the discrete topology, the whole space is the only dense subset. Every non-empty subset of a set X 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 A, B and C of a topological space X with A \subseteq B \subseteq C \subseteq X such that A is dense in B and B is dense in C (in the respective
subspace topology In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced t ...
) then A is also dense in C. 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 f, g : X \to Y into a Hausdorff space Y agree on a dense subset of X then they agree on all of X. For metric spaces there are universal spaces, into which all spaces of given density can be embedded: a metric space of density \alpha is isometric to a subspace of C\left( , 1, \R\right), the space of real continuous functions on the product of \alpha copies of the
unit interval In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analys ...
.


Related notions

A point x of a subset A of a topological space X is called a limit point of A (in X) if every neighbourhood of x also contains a point of A other than x itself, and an isolated point of A otherwise. A subset without isolated points is said to be dense-in-itself. A subset A of a topological space X is called nowhere dense (in X) if there is no neighborhood in X on which A 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 X, a subset A of X that can be expressed as the union of countably many nowhere dense subsets of X 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 X as a dense subset of a
compact space In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
is called a
compactification Compactification may refer to: * Compactification (mathematics), making a topological space compact * Compactification (physics), the "curling up" of extra dimensions in string theory See also * Compaction (disambiguation) {{disambiguation ...
of X. A linear operator between topological vector spaces X and Y is said to be
densely defined In mathematics – specifically, in operator theory – a densely defined operator or partially defined operator is a type of partially defined function. In a topological sense, it is a linear operator that is defined "almost everywhere". ...
if its domain is a dense subset of X and if its range is contained within Y. See also Continuous linear extension. A topological space X is
hyperconnected In the mathematical field of topology, a hyperconnected space or irreducible space is a topological space ''X'' that cannot be written as the union of two proper closed sets (whether disjoint or non-disjoint). The name ''irreducible space'' is pre ...
if and only if every nonempty open set is dense in X. A topological space is submaximal if and only if every dense subset is open. If \left(X, d_X\right) is a metric space, then a non-empty subset Y is said to be \varepsilon-dense if \forall x \in X, \; \exists y \in Y \text d_X(x, y) \leq \varepsilon. One can then show that D is dense in \left(X, d_X\right) if and only if it is ε-dense for every \varepsilon > 0.


See also

* * *


References

proofs


General references

* * * * * * {{DEFAULTSORT:Dense Set General topology