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,
A
X
X
A
X
The of a topological space
X
X.
A subset
A
X
X
X
A
X
A
X
X.
\operatorname{cl}XA=X.
A
\operatorname{int}X(X\setminusA)=\varnothing.
X
A
A.
x\inX,
U
x
A;
U\capA ≠ \varnothing.
A
X.
and if
l{B}
X
x\inX,
B\inl{B}
x
A.
A
B\inl{B}.
An alternative definition of dense set in the case of metric spaces is the following. When the topology of
X
\overline{A}
A
X
A
A
Then
A
X
If
\left\{Un\right\}
X,
X.
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.[1] The empty set is a dense subset of itself. But every dense subset of a non-empty space must also be non-empty.
[a,b]
C[a,b]
[a,b],
Every metric space is dense in its completion.
Every topological space is a dense subset of itself. For a set
X
X
Denseness is transitive: Given three subsets
A,B
C
X
A\subseteqB\subseteqC\subseteqX
A
B
B
C
A
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\toY
Y
X
X.
For metric spaces there are universal spaces, into which all spaces of given density can be embedded: a metric space of density
\alpha
C\left([0,1]\alpha,\R\right),
\alpha
A point
x
A
X
A
X
x
A
x
A
A subset
A
X
X
X
A
X,
A
X
X
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
X.
A linear operator between topological vector spaces
X
Y
X
Y.
A topological space
X
X.
If
\left(X,dX\right)
Y
\varepsilon
One can then show that
D
\left(X,dX\right)
\varepsilon>0.
proofs
A
B
X.
X=\varnothing
A\capB
X
U
X,
U\cap(A\capB)
A
X
U
X,
U\capA
U\capA
X
B
X,
U\capA\capB
\blacksquare