Dense subset

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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 .


  • 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 .


  • 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


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 ).