Dense set
Subset whose closure is the whole space
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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 – Partial order where for every two distinct elements have another element between them Pages displaying wikidata descriptions as a fallback
- 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 . Mineola, N.Y. : Dover Publications . ISBN 978-0-486-43479-7 . OCLC 115240 .