Density (mathematics)

See also: Density (homonymy)

In Topologie, the concept of density of a Sous-ensemble has of a topological Espace X makes it possible to reflect the idea that for any point X of X one can find a point of has which is as close to X as possible.

Either X a topological Space and has a Sous-ensemble of X . has is dense in X so for any element X of X , all Voisinage of X contains at least an element of has .

In an equivalent way, has is dense in X if the only subset closed of X container has is X itself. I.e. the adherence of has is X , or that the interior of the complement of has is empty.

In the case of a metric Space, it is possible to use the following definition: the unit has in a metric space X is dense if any element X of X is the limiting of a continuation of elements of has .

Not dense

A point X of X is dense if the unit { X } is dense in X .

Examples

Any space topological is dense in itself.
the whole of the rational numbers and the irrational numbers is dense in the whole of the real numbers provided with usual topology.
a metric space M is dense in its completion γM .
a separable Espace is a topological space having a dense subset countable.

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Density (mathematics)

Not dense

Examples

See too

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