Dense set

In the mathematical field 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 the topology. It is also used in many other disciplines of mathematics, such as calculus, functional analysis and numerical analysis, for example, in the approximation of continuous functions by polynomials.

It is said of a subset, they lie dense in a metric space, if you can be approximated accurately by a point from the subset each point of the overall space as desired. Thus, the rational numbers form a dense subset in the set of real numbers. This means that you can be approximated accurately by rational fractions or decimals by finite irrational numbers arbitrarily. More generally one says of a subset, they lie dense in a topological space, if contains every neighborhood of an arbitrary point of always an item.

A special case of this topological term closely follows by applying on ordered sets. A subset of a strictly totally ordered set is called dense ( in ) if there is at all and with one out, so. This special case arises on the order topology and is explained in more detail there. This article discusses the more general topological term.

Definition

Be a topological space. A subset is dense in if and only if one (and hence any ) of the following equivalent statements is true:

• The conclusion of not accord with.
• Each environment in contains a point.

Nowhere dense subset

A complementary approach is that of nowhere dense sets: A set is nowhere dense if and only if the amount in any environment one of its elements is dense. This is equivalent to the fact that the interior of the conclusion of the set is empty, that is the largest and only open set that is included in the financial statements is the empty set.

Properties

• Inclusion:
• Transitivity:
• Conservation under continuous maps: Is dense in and a continuous map, so is dense in.

Which is meant tight in accordance with the subspace topology.

Examples

• The set of rational numbers is dense in the set of real numbers.
• The set of irrational numbers is dense in the set of real numbers.
• The set of polynomials is dense in the set of continuous functions on a compact interval.
• The set of test functions is dense in the set of Lebesgue - integrable functions.
• Let be a subset of a normed space means. If we denote by the closed hull of this set with respect to the standard, is dense in.
• The set of natural numbers is not dense in the set of rational numbers, it is even nowhere dense in.
• 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 it is not nowhere dense because it is dense in what is a neighborhood of zero.