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.
Definition in metric spaces
Given a metric space (such as a normalized space with the metric ).
Then a set is called dense in if one of the following equivalent statements is true:
- For each and every one there is a point , so that is.
- For each and every one there is a point , so that is. In this case, referred to the open ball order of radius .
- For each there is a sequence of points such that is.
- The closed shell of the crowd is the whole space, so .
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 .
- The set of irrational numbers is close to the set of real numbers .
- 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 .
- The set of natural numbers is not close to the set of rational numbers , it is even nowhere close to .
- 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.
- The space of the smooth functions with a compact carrier lies close to the space of the square-integrable functions .
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:
- The ending of corresponds to the superset, so it applies .
- The crowd cuts any non-empty open set, so it's for everyone .
- Each environment in contains a point from .
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 .
properties
- Inclusion: if it is tight in and , so is also in .
- Transitivity: if it is dense in and dense in , it is already dense in .
- Conservation under continuous mapping: if dense is in and a continuous mapping is dense in .
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.
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.
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.
Additional terms
Nowhere dense crowds
A nowhere dense set is a subset of a topological space in which the interior of its closure is empty. So it applies
- .
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 .)
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
- ↑ MI Voitsekhovskii: set Dense . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).