Dense subset

In the mathematical subject of topology , a dense subset of a metric or topological space is a subset of this space with special properties. The term dense subset is defined in its general form in topology. It is also used in many other sub-disciplines of mathematics, such as analysis , functional analysis and numerics , for example in the approximation of continuous functions using polynomials .

A subset is said to be dense in a metric space if one can approximate every point of the total space as precisely as required by a point from the subset. The rational numbers thus form a dense subset of the set of real numbers . This means that irrational numbers can be approximated with arbitrary precision using rational fractions or finite decimal numbers . More generally, it is said of a subset , they lie closely in a topological space , where each environment of an arbitrary point from always one element from contains. ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle A}$ ${\displaystyle X}$ ${\displaystyle x}$ ${\displaystyle X}$ ${\displaystyle A}$

Definition in metric spaces

Given a metric space (such as a normalized space with the metric ). ${\displaystyle (X,d)}$ ${\displaystyle (X,\|\cdot \|)}$ ${\displaystyle d(x,y)=\|x-y\|}$

Then a set is called dense in if one of the following equivalent statements is true: ${\displaystyle M\subseteq X}$ ${\displaystyle X}$

• For each and every one there is a point , so that is. ${\displaystyle x\in X}$ ${\displaystyle r>0}$ ${\displaystyle y\in M}$ ${\displaystyle d(x,y)<r}$
• For each and every one there is a point , so that is. In this case, referred to the open ball order of radius . ${\displaystyle x\in X}$ ${\displaystyle r>0}$ ${\displaystyle y\in M}$ ${\displaystyle y\in B_{r}(x)}$ ${\displaystyle B_{r}(x)}$ ${\displaystyle x}$ ${\displaystyle r}$
• For each there is a sequence of points such that is. ${\displaystyle x\in X}$ ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle M}$ ${\displaystyle \textstyle \lim _{n\to \infty }x_{n}=x}$
• The closed shell of the crowd is the whole space, so . ${\displaystyle M}$ ${\displaystyle {\overline {M}}=X}$

The above definition by the limit value of a sequence cannot be transferred to general topological spaces. The convergence of sequences has to be generalized for this by the filter convergence or the convergence of networks .

Examples

• The set of rational numbers is close to the set of real numbers . ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {R} }$
• The set of irrational numbers is close to the set of real numbers . ${\displaystyle \mathbb {R} }$
• The set of polynomials is close to the set of continuous functions on a compact interval.
• The set of test functions is close to the set of Lebesgue integrable functions.
• Let be a subset of a space normalized by means . If one denotes the closed envelope of this set with respect to the norm , then lies in . ${\displaystyle M}$ ${\displaystyle \|\cdot \|}$ ${\displaystyle X}$ ${\displaystyle {\overline {M}}}$ ${\displaystyle \|\cdot \|}$ ${\displaystyle M}$ ${\displaystyle {\overline {M}}}$
• The set of natural numbers is not close to the set of rational numbers , it is even nowhere close to . ${\displaystyle \mathbb {N} }$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {Q} }$
• The Cantor set is an uncountable, closed and nowhere dense subset of the real numbers.
• The interval is not dense in the real numbers, but neither is it dense anywhere, because it is dense in what is a neighborhood of zero. ${\displaystyle [-1,1]}$ ${\displaystyle [-1,1]}$
• The space of the smooth functions with a compact carrier lies close to the space of the square-integrable functions . ${\displaystyle C_{c}^{\infty }(\mathbb {R} ^{n})}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle L^{2}(\mathbb {R} ^{n})}$

Definition in topological spaces

A topological space is given . Then a set is dense if and only if one of the following equivalent conditions is met: ${\displaystyle (X,{\mathcal {O}})}$ ${\displaystyle M}$ ${\displaystyle X}$

• The ending of corresponds to the superset, so it applies . ${\displaystyle M}$ ${\displaystyle {\overline {M}}=X}$
• The crowd cuts any non-empty open set, so it's for everyone . ${\displaystyle M}$ ${\displaystyle M\cap O\neq \emptyset }$ ${\displaystyle O\in {\mathcal {O}}}$
• Each environment in contains a point from . ${\displaystyle X}$ ${\displaystyle M}$

A set is called dense in if it is dense with respect to the subspace topology . Sometimes the dense sets in the superset are also called dense everywhere . ${\displaystyle M}$ ${\displaystyle Y\subset X}$ ${\displaystyle {\mathcal {O}}_{Y}}$ ${\displaystyle X}$

properties

• Inclusion: if it is tight in and , so is also in . ${\displaystyle A}$ ${\displaystyle X}$ ${\displaystyle A\subseteq B\subseteq X}$ ${\displaystyle B}$ ${\displaystyle X}$
• Transitivity: if it is dense in and dense in , it is already dense in . ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle Y}$ ${\displaystyle Z}$ ${\displaystyle X}$ ${\displaystyle Z}$
• Conservation under continuous mapping: if dense is in and a continuous mapping is dense in . ${\displaystyle A}$ ${\displaystyle X}$ ${\displaystyle f\colon X\to Y}$ ${\displaystyle f(A)}$ ${\displaystyle f(X)}$

The last property is provided with the subspace topology of ; the concept of the dense subset is then to be understood with reference to this subspace topology. ${\displaystyle f(X)}$ ${\displaystyle Y}$

Linearly ordered sets

A special case of the topological term dense results from its application to ordered sets. A subset of a strictly totally ordered set is dense (in ) if it at all and made with one out there, so . This special case arises from the order topology on and will be explained in more detail there. ${\displaystyle S}$ ${\displaystyle (M,<)}$ ${\displaystyle M}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle M}$ ${\displaystyle x<y}$ ${\displaystyle z}$ ${\displaystyle S}$ ${\displaystyle x<z<y}$ ${\displaystyle M}$

Partially ordered sets

A different topology is common in partially ordered sets used in forcing theory. For a partially ordered set , the sets (for ) form the basis of a topology . A set closely related if and only if for every element of there is an element which satisfies. ${\displaystyle (P,\leq )}$ ${\displaystyle U_{p}:=\{q\in P\mid q\leq p\}}$ ${\displaystyle p\in P}$ ${\displaystyle \tau _{\leq }}$ ${\displaystyle D\subseteq P}$ ${\displaystyle \tau _{\leq }}$ ${\displaystyle p}$ ${\displaystyle P}$ ${\displaystyle d\in D}$ ${\displaystyle d\leq p}$

Additional terms

Nowhere dense crowds

→ Main article : Nowhere dense crowd

A nowhere dense set is a subset of a topological space in which the interior of its closure is empty. So it applies ${\displaystyle A}$

${\displaystyle \operatorname {Int} ({\overline {A}})=\emptyset }$.

Contrary to their name, nowhere are dense sets not the opposite or complement of dense or everywhere dense sets. More precisely, a set is nowhere dense if and only if it is not dense in any (non-empty) open set. Thus, dense sets are never dense anywhere, since they are always dense in the open set . Conversely, however, there are both non-dense sets that are not dense anywhere (like the integers in ) and non-dense sets that are not dense anywhere (like the interval in .) ${\displaystyle X}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle [2,3]}$ ${\displaystyle \mathbb {R} }$

Separable and Polish rooms

A topological space is called a separable space if it contains a countable, dense set. This often facilitates the demonstration, so separable rooms are "easier" to handle. The concept of Polish space is even stronger ; this is a topological space that contains a countable, dense subset and is completely metrizable .

Web links

• Eric W. Weisstein : Dense . In: MathWorld (English).

literature

• Boto von Querenburg : Set theoretical topology . 3. Edition. Springer-Verlag, Berlin Heidelberg New York 2001, ISBN 978-3-540-67790-1 , doi : 10.1007 / 978-3-642-56860-2 .
• Thorsten Camps, Stefan Kühling, Gerhard Rosenberger: Introduction to set-theoretical and algebraic topology (= Berlin study series on mathematics. Vol. 15). Heldermann, Lemgo 2006, ISBN 3-88538-115-X .
• Dirk Werner : Functional Analysis . 7th, corrected and enlarged edition. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21016-7 , doi : 10.1007 / 978-3-642-21017-4 .

Individual evidence

1. ↑ MI Voitsekhovskii: set Dense . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).