what does it mean for a function to be dense

Subset whose closure is the whole space

In topology and related areas of mathematics, a subset A of a topological space X is said to be dense in 10 if every signal of Ten either belongs to A or else is arbitrarily "close" to a member of A — for instance, the rational numbers are a dumbo subset of the real numbers because every existent number either is a rational number or has a rational number arbitrarily close to it (run into Diophantine approximation). Formally, ${\displaystyle A}$ is dense in ${\displaystyle 10}$ if the smallest airtight subset of ${\displaystyle Ten}$ containing ${\displaystyle A}$ is ${\displaystyle X}$ itself.^[1]

The density of a topological infinite ${\displaystyle X}$ is the least cardinality of a dense subset of ${\displaystyle 10.}$

Definition [edit]

A subset ${\displaystyle A}$ of a topological space ${\displaystyle Ten}$ is said to be a dumbo subset of ${\displaystyle X}$ if any of the following equivalent conditions are satisfied:

1. The smallest airtight subset of ${\displaystyle Ten}$ containing ${\displaystyle A}$ is ${\displaystyle X}$ itself.
2. The closure of ${\displaystyle A}$ in ${\displaystyle X}$ is equal to ${\displaystyle Ten.}$ That is, ${\displaystyle \operatorname {cl} _{X}A=X.}$
3. The interior of the complement of ${\displaystyle A}$ is empty. That is, ${\displaystyle \operatorname {int} _{X}(X\setminus A)=\varnothing .}$
4. Every point in ${\displaystyle Ten}$ either belongs to ${\displaystyle A}$ or is a limit point of ${\displaystyle A.}$
5. For every ${\displaystyle x\in X,}$ every neighborhood ${\displaystyle U}$ of ${\displaystyle ten}$ intersects ${\displaystyle A;}$ that is, ${\displaystyle U\cap A\neq \varnothing .}$
6. ${\displaystyle A}$ intersects every not-empty open subset of ${\displaystyle X.}$

and if ${\displaystyle {\mathcal {B}}}$ is a footing of open sets for the topology on ${\displaystyle X}$ and so this list can be extended to include:

7. For every ${\displaystyle x\in X,}$ every basic neighborhood ${\displaystyle B\in {\mathcal {B}}}$ of ${\displaystyle x}$ intersects ${\displaystyle A.}$
8. ${\displaystyle A}$ intersects every non-empty ${\displaystyle B\in {\mathcal {B}}.}$

Density in metric spaces [edit]

An alternative definition of dumbo fix in the case of metric spaces is the following. When the topology of ${\displaystyle X}$ is given by a metric, the closure ${\displaystyle {\overline {A}}}$ of ${\displaystyle A}$ in ${\displaystyle X}$ is the union of ${\displaystyle A}$ and the set up of all limits of sequences of elements in ${\displaystyle A}$ (its limit points),

$${\displaystyle {\overline {A}}=A\cup \left\{\lim _{north\to \infty }a_{n}:a_{n}\in A{\text{ for all }}due north\in \mathbb {N} \right\}}$$

Then ${\displaystyle A}$ is dense in ${\displaystyle X}$ if

$${\displaystyle {\overline {A}}=X.}$$

If ${\displaystyle \left\{U_{n}\right\}}$ is a sequence of dense open sets in a complete metric space, ${\displaystyle X,}$ then ${\textstyle \bigcap _{north=1}^{\infty }U_{northward}}$ is likewise dense in ${\displaystyle 10.}$ This fact is one of the equivalent forms of the Baire category theorem.

Examples [edit]

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 exist 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 dumbo subsets (in particular, two dense subsets may be each other'due south complements), and they need not fifty-fifty be of the same cardinality. Perhaps even more than surprisingly, both the rationals and the irrationals have empty interiors, showing that dumbo sets demand not incorporate whatsoever non-empty open fix. The intersection of 2 dense open subsets of a topological space is once again dense and open.^{[proof 1]} The empty prepare is a dumbo subset of itself. But every dense subset of a non-empty space must too exist non-empty.

By the Weierstrass approximation theorem, any given circuitous-valued continuous office divers on a closed interval ${\displaystyle [a,b]}$ can exist uniformly approximated every bit closely as desired by a polynomial function. In other words, the polynomial functions are dumbo in the infinite ${\displaystyle C[a,b]}$ of continuous complex-valued functions on the interval ${\displaystyle [a,b],}$ equipped with the supremum norm.

Every metric space is dumbo in its completion.

Properties [edit]

Every topological space is a dense subset of itself. For a set ${\displaystyle Ten}$ equipped with the discrete topology, the whole space is the merely dense subset. Every non-empty subset of a fix ${\displaystyle X}$ equipped with the petty topology is dense, and every topology for which every non-empty subset is dense must be trivial.

Denseness is transitive: Given three subsets ${\displaystyle A,B}$ and ${\displaystyle C}$ of a topological space ${\displaystyle X}$ with ${\displaystyle A\subseteq B\subseteq C\subseteq 10}$ such that ${\displaystyle A}$ is dense in ${\displaystyle B}$ and ${\displaystyle B}$ is dumbo in ${\displaystyle C}$ (in the respective subspace topology) then ${\displaystyle A}$ is likewise dense in ${\displaystyle C.}$

The paradigm 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 adamant by their values on dense subsets: if 2 continuous functions ${\displaystyle f,thousand:X\to Y}$ into a Hausdorff space ${\displaystyle Y}$ concord on a dense subset of ${\displaystyle 10}$ then they agree on all of ${\displaystyle 10.}$

For metric spaces there are universal spaces, into which all spaces of given density tin be embedded: a metric space of density ${\displaystyle \blastoff }$ is isometric to a subspace of ${\displaystyle C\left([0,i]^{\alpha },\mathbb {R} \correct),}$ the space of real continuous functions on the production of ${\displaystyle \alpha }$ copies of the unit interval. ^[2]

[edit]

A point ${\displaystyle 10}$ of a subset ${\displaystyle A}$ of a topological infinite ${\displaystyle Ten}$ is called a limit point of ${\displaystyle A}$ (in ${\displaystyle X}$ ) if every neighbourhood of ${\displaystyle x}$ also contains a point of ${\displaystyle A}$ other than ${\displaystyle ten}$ itself, and an isolated point of ${\displaystyle A}$ otherwise. A subset without isolated points is said to be dense-in-itself.

A subset ${\displaystyle A}$ of a topological space ${\displaystyle X}$ is called nowhere dense (in ${\displaystyle X}$ ) if there is no neighborhood in ${\displaystyle 10}$ on which ${\displaystyle A}$ is dumbo. 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 up set. Given a topological space ${\displaystyle X,}$ a subset ${\displaystyle A}$ of ${\displaystyle 10}$ that can be expressed as the wedlock of countably many nowhere dense subsets of ${\displaystyle X}$ is chosen 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 infinite if and merely if the intersection of countably many dense open up sets is ever dense. A topological space is called resolvable if it is the matrimony 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 ${\displaystyle 10}$ as a dense subset of a compact space is called a compactification of ${\displaystyle X.}$

A linear operator between topological vector spaces ${\displaystyle 10}$ and ${\displaystyle Y}$ is said to be densely defined if its domain is a dense subset of ${\displaystyle Ten}$ and if its range is contained within ${\displaystyle Y.}$ Come across also Continuous linear extension.

A topological space ${\displaystyle X}$ is hyperconnected if and only if every nonempty open set up is dense in ${\displaystyle X.}$ A topological space is submaximal if and only if every dense subset is open.

If ${\displaystyle \left(10,d_{Ten}\correct)}$ is a metric space, and then a non-empty subset ${\displaystyle Y}$ is said to exist ${\displaystyle \varepsilon }$ -dense if

$${\displaystyle \forall x\in X,\;\exists y\in Y{\text{ such that }}d_{10}(ten,y)\leq \varepsilon .}$$

One tin then show that ${\displaystyle D}$ is dense in ${\displaystyle \left(X,d_{X}\right)}$ if and but if information technology is ε-dumbo for every ${\displaystyle \varepsilon >0.}$

Run into also [edit]

• Blumberg theorem – Any real function on R admits a continuous restriction on a dense subset of R
• Dumbo order
• Dense (lattice theory)

References [edit]

1. ^ Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, ISBN0-486-68735-X
2. ^ Kleiber, Martin; Pervin, William J. (1969). "A generalized Banach-Mazur theorem". Bull. Austral. Math. Soc. 1 (2): 169–173. doi:x.1017/S0004972700041411.

proofs

General references [edit]

• Nicolas Bourbaki (1989) [1971]. General Topology, Chapters 1–4. Elements of Mathematics. Springer-Verlag. ISBN3-540-64241-two.
• Bourbaki, Nicolas (1989) [1966]. General Topology: Capacity i–4 [Topologie Générale]. Éléments de mathématique. Berlin New York: Springer Science & Business Media. ISBN978-3-540-64241-1. OCLC 18588129.
• Dixmier, Jacques (1984). General Topology. Undergraduate Texts in Mathematics. Translated by Berberian, S. Grand. New York: Springer-Verlag. ISBN978-0-387-90972-i. OCLC 10277303.
• Munkres, James R. (2000). Topology (2nd ed.). Upper Saddle River, NJ: Prentice Hall, Inc. ISBN978-0-thirteen-181629-ix. OCLC 42683260.
• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of 1978 ed.), Berlin, New York: Springer-Verlag, ISBN978-0-486-68735-3, MR 0507446
• Willard, Stephen (2004) [1970]. General Topology (First ed.). Mineola, Northward.Y.: Dover Publications. ISBN978-0-486-43479-7. OCLC 115240.

smithocks1947.blogspot.com

Source: https://en.wikipedia.org/wiki/Dense_set

Share this post