Together nowhere dense

In Topologie, a Ensemble is nowhere dense if it satisfies the properties opposite of the concept of density. Intuitively, a Sous-ensemble has of a topological Espace X is nowhere dense in X so almost no point of X cannot be “approximate” by points of has .

Definition

Either X a topological Space and has a Sous-ensemble of X . has is nowhere dense in X if the interior of the adherence of has is empty. Such a part has is also qualified of rare .

The order of the definition is important: it is possible to find subsets dense of which the adherence of the empty interior east (it is the case of the rational numbers as a whole of the real numbers).

Properties

Any subset of a unit nowhere dense is nowhere dense and the union of a number finished of units nowhere dense is nowhere dense. On the other hand, the union of a countable number of units nowhere dense is not inevitably nowhere dense.

Examples

the whole of the integers is nowhere dense in the whole of the real numbers provided with usual topology.

Measure of Lebesgue positive

A unit nowhere dense is not necessarily negligible. For example, if X is the interval, it is not only possible to find a subset dense of null Mesure of Lebesgue (like the whole of the rational numbers), but it is also possible to have a subset nowhere dense of positive measurement of Lebesgue.

See too

Internal bonds

Density (mathematics)

External bonds

'' dense Some nowhere sets with positive measure ''

Category: General topology

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